ANNALS OF DISCRETE MATHEMATICS
Managing Editor Peter L. HAMMER, University of Waterloo, Ont., Canada Advisory Editors C. BERGE, UniversitC de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. G.-C. ROTA, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM NEW YORK. OXFORD
ANNALS OF DISCRETE MATHEMATICS
TOPICS ON STEINER SYSTEMS
Edited by C.C. LINDNER Auburn University, Auburn, AL 36830, U.S.A.
and A. ROSA McMaste r University , Hamilton, Ont ., Canada.
1980
NORTH-€1OLLAND PUDLISHINC COMPANY
-
AMSTERDAM NEW YORK OXFORD
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PREFACE
The concept of a Steiner Triple System is so simple that any reasonably intelligent grade school student could understand it. A mathematician not familiar with the notion and its history, may well have concluded that the Kirkman paper of 1847, which settled the existence question, said all there was to say about this “puzzle”. Instead of this we find that in the intervening 132 years a large and impressive literature has grown up. The subject is still a lively one and there are no signs of a diminished interest in it. The interest is drawn from several sources. birstly, there is a large body of mathematics which has been used in the study of Steiner Triple Systems and their natural extensions. What immediately comes to mind are such things as cyclotomy and Galois Field Theory and other areas of algebraic numbers theory. Secondly, there is a connection between problems in combinatorial systems and other areas of mathematics. For instance, the work of Trevor Evans has shown that the problems of embedding partial systems in complete systems is equivalent to the solution of a word problem in certain algebras. One of the equivalents of the axiom of choice is the existence of Steincr Triple Systems on certain sets. Furthermore, the general Steiner Systcms have an interesting algebraic structure in themselves and their automorphisd groups are an attraction to many mathematicians particularly those interested in sporadic simple groups. Again, they have been useful in settling problems in other areas of mathematics, notably in lattice theory and universal algebra. It is of some interest to speculate on why such apparently diverse areas of mathematics are related to each other and why Steiner Systems play such an important role. How can such rich connections occur if mathematicians merely play a meaningless game with arbitrary axiom systems. Of course, mathematicians may concede that in form what they do may be described as a meaningless game. but in actuality what they achieve is a rich structure far removed from the sterility implied by the formal description. There is a mystique that mathematicians have so far avoided sterility because their axiom systems are not arbitrary or random, but are derived initially by abstracting from occurrences in “nature”. This mystique, in fact, has a sound rational basis. When a mathematical system originates in a concrete natural model the mathematician has available to him the ability to construct examples and counterexamples which guide him to conjectures on which he builds a cohesive structure. In fact, the mathematician operates as if he were an experimental scientist developing his theory from observations. For example, if one examines the way in which theorems in the theory of numbers were first conjectured by an experimental examination of the integers, the process bears a striking resemblance to the way in which a physical theory is built up from V
\i
Preface
experimental observation. And number theory has not yet lost any of its vitality. The same can be said from the present state of Steiner systems. Since Steiner Systems have a rich and cohesive structure let us examine their roots. The prime source of any combinatorial structure is geometry. The notion of a finite plane was derived from the “physical” planes of experience. It turns out that the Steiner Triple System of order 7 is the smallest finite projective plane while the system of order 9 is an afine plane of the next higher size. The use of the Galois Field for construction of systems thus becomes natural. The more general Steiner Systems which are associated with the Mathieu groups also have arisen naturally in more than one way. While attempts of mathematicians to use such systems in the construction of further quadruply and quintuply transitive groups have failed, the analogous use of doubly and triply transitive groups to construct cornbinatorial designs has been highly successful. This long preface to a volume on Steiner Systems was motivated by the depth and breadth of the papers included. Virtually no area of activity was overlooked. The volume gives an excellent survey of the state of the art and the direction in which research is moving. It is a most welcome addition to the literature and the editors are to be congratulated for their selection of participants, and the participants are to be congratulated for their contributions. N.S. Mendelsohn
FROM THE EDITORS In 1844 W.S.B. Woolhouse asked: for which integers 1, k, and u is it possible to construct a collection B of k-subsets of a set S of size u so that every t-subset of S is contained in exactly one member of B? In today’s terminology such a pair ( S , B )is called a Steiner system S(?,k, u ) . Three years later in 1847 the Rev. T.P. Kirkman obtained necessary and sufficient conditions for a Steiner system S(2,3, u ) to exist. One might wonder why, if the systems S(t, k, u ) were introduced by Woolhouse and the first major result was obtained by Kirkman, are they called “Steiner systems”. In 1853 J. Steiner unaware of the papers by Woolhouse and Kirkman, asked for the existence of the systems S(t, t + 1, u ) . The fact that the systems S(t, k, u ) still carry the commonly accepted name “Steiner systems” (a term coined by E. Witt) is most certainly due to the fact that writers in the late nineteenth and early twentieth century, just as Steiner, were unaware of the papers by Woolhouse and Kirkman and referred to the paper by Steiner. Regardless of the origin of the name “Steiner systems”, it is safe to say that the subject has grown to the point where today it occupies a significant position in combinatorial mathematics. To-date over 700 papers on Steiner systems have been written, the majority in the past 25 years. Obviously it would take an encyclopedia to even begin surveying all the aspects of Steiner systems which have been studied since Woolhouse’s day, and this volume is no such attempt. What we have tried to do here is bring together a collection of articles (both survey and research) which reflect a large (even if not exhaustive) portion of current research on Steiner systems. We would like to thank all the contributors for responding to our invitation and for their contribution to this volume. All articles were thoroughly refereed, and we would also like to extend our thanks to the referees for their effort.
C.C. Lindner and A. Rosa
Vii
CONTENTS
Preface (N.S. MENDFI.SOHN)
V
From t h e Editors (C.C. LINDNER and A. ROSA)
vii 1
1. Algebraic aspects of Steiner systems
B. GANTER and H. WERNER, Co-ordinatizing Steiner systems R.W. QUACKENRUSH, Algebraic speculations about Steiner systems L. BARN.Almost all Steiner triple systems are asymmetric
3 25 37
11. Steiner systems with higher value of r
41
P . J . CAVFRO\,External results and configuration theorems for Steiner systems R.H.F. D E N N I S I O N . The problem of the higher values of t
65
43
111. Steiner systems with given properties, collections of Steiner systems,
and their relationship to other combinstorial conligurstions
71
A.J.W. HIi:rm and L. TEIRLINCK, Dimension in Steiner triple systems E. MFNDF.I.SOHN. Perpendicular arrays of triple systems R.C. MUI.LIN and S.A. V A N S T O N E , Steiner systems and Room squares K.T. PriFi.ps, A survey of derived triple systems '4. ROSA.Intersection properties of Steiner systems
73 89 95 105 115
IV. Resolvability and embedding problems for Steiner systems
129
J. HA
1..
On identifying PG(3.2) and t h e complete 3-design on
seven points A. HARTMAN. A survey on resolvable quadruple systems M. Limos. Projective embeddings of small "Steiner Triple systems" C'.C'. L .I'U'I)NER. A survey of embedding theorems for Steiner systems D . E . WOOI.RRIGITI. O n the size of partial parallel classes in Steiner systems
203
V. Isomorphism problems and enumeration of Steiner systems
213
M.J.COI.HOUKN and R.A. MATHON.On cyclic Steiner ?-designs
215
R . H . F . DENNISION. Non-isomorphic reverse Steiner triple systems o f order 19
255
...
v111
131 143 151 175
Contents
ix
L.P. PETRENJUK and A.J. PETRENJUK, An enumeration method for nonisomorphic combinatorial designs K.T. PHELPS, On cyclic Steiner systems S(3,4,20) I. DIENER, On cyclic Steiner systems S(3,4,22)
265 277 30 1
VI. Bibliography and survey of Steiner systems
315
J . DOYENand A. ROSA, An updated bibliography and survey of Steiner systems
317
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PART I
ALGEBRAIC ASPE<;TS OF STEINER SYSTEMS
This Page Intentionally Left Blank
Annals of Discrete Mathematics 7 (1980) 3-24 @ North-Holland Publishing Company
CO-ORDINATIZING STEINER SYSTEMS Bernhard GANTER FB4, AGI, Technische Hochschule, 6100 Darmstadr, West Germany
Heinrich WERNER FB17. Mathernatik, Gesamthochschule. Heinrich-Plert-Str. 40, 3500 Kassel, West Germany
This article is neither a research paper nor a survey on the algebraic methods applicable to Steiner Systems: it is merely a collection of folklore. We describe the techniques used to co-ordinatize Steiner Systems by algebras, with the (strong) restriction that only those co-ordinatizations are discussed which lead to algebraic “varieties”. We have not attempted the impossible task of assigning authors to definitions and results. The main contributors to the topics treated are R.H. Bruck, T. Evans, R.W. Quackenbush, and S.K. Stein, but many others, among them A.A. Albert, H.-J. Arnold, E . Mendelsohn, N.S. Mendelsohn, R. Metz, J.M. Osborn, N.K. Pukharev, L. Szamkol’owicz, should also be mentioned. We also d o not feel responsible for the names given to the co-ordinatizing algebras. We have adopted T. Evans’ “SQS-skeins” as well as R.W. Quackenbush’s “near vector spaces” although the first sounds awful (to us) and the second has been used in the literature with at least two other meanings. We assume that the reader is familiar with the definition of a Steiner System; by a Steiner System of type (t, k ) or a (t, k)-Steiner System we mean a Steiner System S(r, k , u ) for some u.
I
S L O O P S
L e t ( S ; o , e ) b e a sloop, i.e. a groupoid with a neutral element e s a t i s f y i n g t h e identities x o e = x x o y = Y O x 0 (x 0 y)
.-
X
=
y.
{ { x , y , x o y } l x , y € P , x z y }.
Let (P,B) b e a S t e i n e r T r i p l e S y st em . D e f i n e on
S : = P lj { e } a b i n a r y o p e r a t i o n o by x o x := e , e o e : = e x o e : = e o x : = x x o y : = 3 r d p o i n t on t h e block through x,y
for all
x,y
L
P, x
(S;o,e) i s a sloop.
L
y.
A
6. G a n f e r . H . Werner
Let
(P;o) be a squag, i . e . a groupoid satisfying
Let (P,B) be a Steiner Triple System.
the identities x o x = x
Define on P a binary operation o by x o x : = x x o y : = 3rd point on the block through x,y for xsy ' y.
x o y = Y O X x 0 (x 0 y) = y. Define
B
:=
[(x,y,xoy}lx,ycP,
'
XLY}.
is a Steiner Triple
(P,BJ
(P;o)
9
is a squag.
System.
N E A R
V E C T O R
S P A C E S
Let ( Q ; @ , O ) be a near vector space over GF(31, i.e. a groupoid with a neutral element 0 satisfying the identities o o x = x x o y = y o x ( x o x ) o y = x o ( x o Y ) x 0 (x 0 (x 0 Y)) = y.
O V E R
GF(3)
Let (P,B) be a (2,4)-Steiner System. Let Q : = P x {1,21 C {O]. For each block b E B choose a bijection @b:(bx{l ,2)u10I)-GF(3)* such that @b(0) = ( 0 , O ) and @h((X,2))
=
2@h((X91)).
~
On Q define a binary operati Define [XI : = tx, x 0 x 1 , on 0 by P : = ( I x I I x e Q , x L 01, @ y :=@: (@b(x) + @b(y)) B : = 1 1 . r x l , ~ y l , r x ~ 1 , c x o ~ x o y ~for l l all x,y c Q and b c B x,). c Q ' \ ( O I , x [Yl). for which @,(x) and @,(y) are defined. (P,B)
is a (2,4)-Steiner
System. IQI = Z I P 1
(Q;O,O) is a near vector space over C F ( 3 ) . +
1
Co-ordinatizing Steiner systems
S T E I N
Q U A S I G R O U P S
Let (P;o) be a Stein quasigroup, i.e.a groupoid satisfying the identities x o x = x
(x o y ) o y = y o x
(y
0
5
x) 0 y = x .
Let (P,B) be a (2,4)-Steiner System. On {0,1,2,3} define *I0 1 2 a a binary operation * 0 0 2 3 1 by means of the table in the margin.
,
For every block bcB choose a bijection $,,:b----){O,l ,2,3}. 3n
P
define an operation o b * $b(y)) Y : = $%($b(') for all x,ycP, bcB for which eb(x), gb(y) are defined. (P;o)
(P,B) is a (2,4)-Steiner System.
is a Stein quasigroup. I
~~~
N E A R
V E C T O R
S P A C E S
O V E R
GF(q)
X denotes (the scalar multiplication with) some fixed primitive element of GF(q). GF(q)* := GF(q)\{O).
Let (Q;+,X,O) be a near vector space over GF(q), i.e. an algebra satisfying all identities in two variables that hold in vector spaces over GF(q)
.
Define [XI
P B
:= :=
= I ,. . . , q - 1 1 , Q, x * 01, {{[x+,~"y~ln= I , ...,q- 1 1 1 X,YEQ, XLY, x*O*y}.
:=
{X"xln
ICxIlx
E
(P,B) is a (Z,q+l)-Steiner System.
Let
(P,B) Sys tem.
be a (Z,q+l)-Steiner
Q : = P x GF(q)*\{O). For each B choose a bijection $b: ((bxGF(q)*) uIO1)-GF(q) such that bb(0) = ( 0 , O ) and, for each pcb, $b induces a linear mapping from ({plxGF(q)*u{Ol) into GF(q) 2 Let
b
E
',
.
Q, define Y := $b(y)) : = 6i(x$b(x)) for all x , y ~ Q ,bEB for which ) defined. $b(x), $ ~ ~ ( y are
On
+
$2($b(x)
+
(Q;+,X,O) is a near vector space over GF(q).
S . Ganrer. H . Werner
h
A L G E B R A S
B L O C K
F
Let
=
be a f i n i t e n e a r f i e l d and
(F;+,-,.,O,l)
F
element which g e n e r a t e s On
*
define a binary operation
F
Let
as a ring.(iFI=:q
(f;o)
which a r c t r u e i n
a bijection
P
On
:'x,y, !x,y
%(y)
b e a n e a r boo-
l e a n a l g e b r a , i.e. a n a l g e b r a s a t i s f y i n g a l l i d e n t i t i e s i n two v a r i a b l e s which a r e t r u e i n boolean a l g e b r a s . Lct I) b e a s u b s e t o f
I'
1''
=
Q\lO,l
Let P ' b e a s e t w i t h P ' n P !PI = I P ' I , and l e t ':P+P' be a b i j e c t i o n . Q := P u P '
=
6,
u {O,l).
a, On
~
~
Q
define binary opcra-
~~
For x , y ~ l ' , x ~ y ,l c t < x , y > d e n o t e t h e s u h a l y c b r a g e n e r a t e d hy ix,y!. Thr set <x,y>!lP\(x,y! e i t h e r c o n s i s t s of a s i n g l c e l e ment 2 o r c o n t a i n s a u n i q u e c l e mcilt 2 w h i c h i s t h e j o i n o f two atoms o r ' . ~ , y > .
(P,B)
A L G E B R A S
Let ( P , B ) b e a S t e i n e r ' T r i p l e System.
Let
i
= Q,{O,lj. ~~~~~
are defined.
(F;*).
B O O L E A K
N E A R
1''
choose
(P;o) i s a block algebra over
J
B
0 y := %l(%(x)*%(~)) f o r a l l x , y E P f o r which
i s a ( , i , q ) - S t e i n e r Systen
such t h a t
t
define a binary operation
%(x),
Let ( Q ; v , " , ' , O , l )
b
h:b--pF.
x
o by
g r o u p o i d g c n c r a t e d by x a n d y .
I'
(x-y).a.
+
(F;*).
~ x , y >denotes t h e sub-
(P,B)
F \ ( 0 , 1 ) an
be a (2,q)-Steiner
For e v e r y b l o c k
i d e n t i t i e s i n two v a r i a b l e s
rilierc
(P,B)
E
System.
i . c . a groupoid s a t i s f y i n g a l l
Define R : =
x * y := y
by
Let
he a b l o c k a l g e b r a
(F;*),
over
a 2).
>
tions
V , A
i n s u c h a way t h a t
f o r every block
h c R
( b u b ' u {O,ll;v,~,',O,l) i s a boolean algebra.
( Q ; v , ~ , ' , o , l ) is a
is a Steiner Triple
near boolean a l g e b r a .
System. 1Q1
=
ZIP1
+
2
Co-ordinatizing Sfeiner systems
S Q S
-
Let ( P ; q ) b e a n S Q S - s k e in , i.e. q is a t e r n a r y operation on P s a t i s f y i n g t h e i d e n t i t i e s
Y q(x,y,z) = q(x,z,y) q ( x , y , q ( x , y , z ) ) = 2. q(x,x,y)
S K E I N S Let ( P ,B) b e a S t e i n e r Q u a d r u p l e S y st em .
=
=
7
q(y,x,z)
D e f i n e on
P
a ternary
operation
q
by
q(x,x,y)
:=
q(x,y,x)
q(y,x.x) q(x,y,z)
:= Y
:=
4 t h p o i n t on t h e block through X , Y , a n d 2, f o r a l l x,y,zcP, XLYLZZX.
i s a S t e i n e r Quadruple
(P,B)
(P;q)
:=
i s a n S O S - sk ei n .
S y s t em
1. Co-ordinatizations With the abbreviation [x, y]" := x, [ x , y ] " + ' : = [ x , y]"oy,
n=0,1,2, ...
we can, for example, give defining identities for each variety of block algebras where t h e nearfield 9 is a field and t h e element a is primitive: x o x = x, [x, y]" = [y.
[x, yp--1 = x,
XI" whenever
1 s n, rn < q - 1 are such that a" + a"'
=
1.
We shall call such a variety a variety of block algebras ouer GF(q) although there is danger of confusion: The identities depend on t h e choice of the element a. There are, for example, two different varieties of block algebras over GF(S), corresponding to t h e primitivc elements a = 2 and a = 3:
(((x 0 y )o Y ) 0 Y )o Y = x.
For a = 3 the operation
0
is commutative, for a
=2
it is not.
B . Canter. H . Werner
8
The identities obtained in this way are not necessarily independent. Of course there are always several sets of identities defining a particular variety. N.S. Mendelsohn proved that the variety of all sloops can be defined by the single identity x ~ ( ( y ~ y ) ~ ~2 , ) ~ x ) =
D. Johnson has shown that one of the varieties of block algebras over GF(11) is definable by the two identities (y0x)ox = x o y
and
((xoy)o(yox))o(xoy)= y.
Each of t h e preceding examples establishes a correspondence between all Steiner Systems of a fixed type (1, k ) and a class of algebras' defined by a set of identities. Every such algebra co-ordinatizes a Steiner System of the respective type, and vice versa, each (t, k)-Steiner System is the underlying system of some (sometimes many) algebras in the class. The co-ordinatization is unique only for sloops, squags, and SQS-skeins (for squags and SQS-skeins it is even functorial). Except for near boolean algebras every co-ordinatizating algebra can be obtained from a Steiner System via the described construction. Note that squags and Stein quasigroups are special block algebras ( q = 3, q = 4) whereas sloops are near vector spaces over GF(2). We have already mentioned that all the classes of algebras defined in this paragraph are equational classes or uariefies, i.e. they are defined by identities (For block algebras we have an equational class for every particular choice of the element a E 9.). Varieties of algebras are closed under the formation of subalgebras (see Section 2), Cartesian producrs (Section 3). and quotient algebras (Sections 4-5). In some of the examples the identities have only implicitly been given, e.g. for block algebras. But for every choice of 9 and a E 3 an explicit set of (finitely many) identities defining t h e respective variety can be given.
2. Subalgebras A subspace of a Steiner System (P, B ) is a set S c P which is closed under forming blocks. A subalgebra of an algebra ( A ;F ) is a set S c A which is closed under the fundamental operations.
-
' W e LISC the word "algebra" as a generic term for algebraic structures like quasigroups. vector\pace\. lattices. ctc More precisely: An n-ary operarion f on a set A is a mapping f : A" A. An afRehrtr is a pair ( A where F = (jl, f,, . . .) is a family of ("fundamental") operations on A. The q u e n c e In,. P I ? . . . . ) of the arities of these operations is the type of the algebra ( A ; F ) . We shall consider o n l y classes in which all algebras have the same type. A groupoid ( A ; e.g. is of type (2). it has a single hinary ( = 2-ary) fundamental operation. In this case it is common to write xoy instead of ' I H . y 1 Another example: Near boolean algebras are of type ( 2 , 2 , I , 0.0) (constants are treated as nullarv operations).
;m.
0 )
9
Co-ordinatizing Steiner systems
If (P,B) is a Steiner System of type (t, k), then a subset S E P is a subspace iff JbflSlst j ~ G S .
VbEB
Trivially each subset of cardinality less than t is a subspace, and so is each block and the set P itself. If S is a subspace of (P,B), then (S, B f l P(S)) is a subsystem of (P,B). A subset S is a subalgebra of an algebra (A; F) iff
n,, . . . , x,,,ES j fi(x,,..., x , , ) E S .
V f i ~ F v x,..., ,
For example, a subset S E A is a subalgebra (subgroupoid) of a groupoid ( A; iff s, xoy E s. Varieties are closed under the formation of subalgebras: Every subalgebra of a squag is a squag, of a sloop is a sloop, etc. There is a 1-1-correspondence between the subalgebras of a coordinatizing algebra and the subspaces of the underlying Steiner System, except for near boolean algebras where some of the blocks may not correspond to subalgebras. A subalgebra (subspace) S is generated by a set G (in symbols: (G) = S ) if S is the smallest subalgebra (subspace) containing G. S is called n-generated if it is generated by an n-element set and strictly n-generated if, in addition, it is not generated by a set of smaller cardinality. The co-ordinatizing varieties can be characterized by their ?-generated algebras: An algebra (of appropriate type) is a 0)
vx, y E
Near vector space Every 2-generated subalgebra is a vector space over GF(q) over GF(q) Block algebra over ( F ;
Every 2-generated subalgebra is isomorphic to ( F ;
Near boolean algebra
Every 2-generated subalgebra is boolean
SQS-skein
Every 3-generated subalgebra is isomorphic to the 4-element SQS-skein
0)
0)
In particular, a groupoid is a squag iff every two distinct elements generate a 3-element subgroupoid isomorphic to the one displayed below.
I
x
Y
XOY
The 3-element squag
Moreover, a groupoid is a sloop iff every strictly 2-generated subgroupoid is isomorphic to the Klein four group A,.
I0
8 . Canter.
H. Werner
The four element SQS-skein (sometimes called Swierdzkowski-algebra) is Q,:=(A,;x+V+z). The next table shows the connection between “small” subalgebras and their underlying subspaces:
II
algebra
n
I
i sloop
1 2
2 4
point hlock
1
1 3
point hlock
j squag I
,
1 nearvector
1
j
&
:
1
3
nearboolcan algehre
I
4
2
X o r I6
I 2
3
SOS-skein
I
1
aleehra
j
I
point hlock
space
rbGz1
cardinality of a strictly n-generated subalgebra underlying suhspacc
hlock
1
pmnr block or S(Z. 3 . 7 ) point 3 points
3. Products The Cartesian (or direct) product of two algebras ( A :F) and ( 5 ;F) is the algebra ( A :F ) x
( B ;F ):= ( A x B ; F ) ,
where the operations f, E F are defined on A x R “componentwise” by
For example. the direct product of two groupoids ( A; 0) and ( B ;0) is t h e groupoid ( A x B : . ) detined by
Cartesian products are defined only for algebras of the same type. Varicties are closed under products: The direct product of sloops is a sloop, etc. The cardinality of the direct product is the product of the cardinalities of the factors. Thus t h e existence of a single finite algebra with at least two elements in a iarietv 3’guarantees the existence of infinitely many finite algebras in s‘.
Co-ordinatizing Sreiner sysrems
11
Fig. 1. The direct square of the 3-element squag.
An analogous construction for Steiner Systems in general does not exist: A direct product of an S ( 2 , 6 , 6 ) with itself would be an S ( 2 , 6 , 3 6 ) ; the parameter 25 = 5 . 5 is not even admissible for (4,5)-Systems.’ But in the cases where we have co-ordinatizations (that is, for ( t , k ) = ( 2 , q ) , ( 2 , q + l ) , or (3,4)), the Cartesian product of the co-ordinatizing algebras can be used to construct new Steiner Systems from old. The procedure is as follows: We start with two Steiner Systems ( P , , B , ) and ( P 2 , B2) of appropriate type (t, k ) , co-ordinatize (in the same variety, of course), and obtain the new system as the underlying of the Cartesian product of the two co-ordinatizations. (Pi, B,)
(PZ.B’)
I
co-ordinatization
factor systems
I
co-ordinatization
I
I
factors (algebras)
\
/
product algebra
underlying system
(P, B)
product system
As the Steiner System and the co-ordinatizing algebra usually do not have the same number of elements, the cardinality of the product system is not always the product of the cardinalities of the factor systems. The following table shows which ’The authors wonder if there is a reasonable way to turn the class of Steiner Systems of type (r, k ) into a category with products for some ( r , k ) # ( 2 . 3 ) , ( r , k ) # (3.4).
B. Ganter. H. Werner
12
cardinalities are obtained forming the product of an S(t, k, u ) and an S(r, k, w):
I
1 variety sloops squags near vector spaces block algebras ncar boolean algebras SOS-skeins
(1.
k)
(2.3) (2.3) ( 2 , q + 1) (2.4) (2, 3) (3.4)
cardinality of the product system
(u + I ) ( w + 1 ) - 1 uw
(q - 1)uw + u + w uw
2(u + I ) ( w + 1 ) - 1 uw
Of course, all these constructions can be performed without any coordinatization, in fact, most of them occur in Witt’s paper. Many of the attempts to co-ordinatize Steiner Systems were motivated by the hope to find further product constructions (the admissible values for (5,6)-Systems e.g. are closed under products), but n o new values were found in this way (so far). Nevertheless, in the algebraic setting the known product constructions are much more natural and easier to handle.
Fig. 2. The direct product of the 4-element sloop with the 2-element sloop.
We list their elementary properties: The product of subalgebras is always a subalgebra of the product. Furthermore a product of an algebra ( A; F ) with a one-element algebra is trivially isomorphic to ( A :F ) . Thus if one of t h e factors contains a one-element subalgebra then the other factor is (isomorphic to) a subalgebra of the product. With the exception of near boolean algebras all co-ordinatizing algebras have one-element subalgebras. So for each co-ordinatization except possibly for near boolean algebras, the factor systems are subsystems of the product system. SQS-skeins and block algebras (in particular, squags) are idernporent i.e. each singleton is a subalgebra. In a direct product of two such algebras t h e set of all pairs with fixed second component is a subspace whose subsystem is isomorphic to t h e first factor system, and the analogous situation holds for the second factor. The product system thus can in the case of idempotent co-ordinatization be partitioned i n t o subsystems isomorphic to a given factor.
Co-ordinatizing Steiner systems
13
As points and blocks are subspaces, they lead to subalgebras of the coordinatizations (with exceptions again in the case of near boolean algebras) and their products cause numerous subsystems of the product system. The following table shows which systems occur:
I variety
factor 1 factor 2
subsystem of the product
I
point point block
point block block
projective plane projective 3-space
point
point
point
spaces
point point block
point block block
block projective plane projective 3-space
block algebras
point point block
point block block
point block f i n e plane (over the nearfield 9 )
sloops
squags affine olane
point point point 2 points SQS-skeins point block 2 points 2 points 2 points block block block
point
2 points block block S(3,4,8) S(3.4, 16)(affine)
It seems funny that for near vector spaces the projective spaces can be generated “from nothing”; they are just the Cartesian powers of a single point. The same happens for SQS-skeins if we start with 2 points. We close this chapter with an example of a structure theorem for direct products which goes back to Birkhoff:
Theorem. If “Ir is a variety of algebras such that each member of “Ir has permutable congruences and a singleton subalgebra, then each finite algebra in Q has a decomposition into a direct product of directly indecomposable factors which is unique up to isomorphism and permutations of the factors. As we shall see in Section 5 this theorem applies to all co-ordinatizing varieties except to near boolean algebras (which do not have singleton subalgebras). But it can be generalized to near boolean algebras, thus all these products have the unique factorization property. 4. Morphisms A morphism between two Steiner Systems is a mapping which preserves subspaces.
8.Canter, H . Werner
14
A hornotnorphism between two algebras is a mapping which preserves the
fundamental operations. More precisely: A mapping q : PI
-
P2
is a morphism from the Steiner System (f,, €3,) to the Steiner System ( P 2 ,B 2 ) if the systems are of the same type and the image of each subspace of (PI,B , ) is a subspace of (P2.Bz). (An equivalent condition is that the image of each block is a block or has less than r elements). A mapping
q:A-+B is a homomorphism from the algebra ( A ;F ) to the algebra (€3; F ) if t l f , ~ F V a ,. . . . u , E A v ( f i ( a l s .. .. 4,,))=f,(cp(a1). . . ..v(a,,)).
Homomorphisms preserve subalgebras. The combinatorial interpretation of the homomorphisms is quite different for the co-ordinatizing varieties. We discuss the cases separately, excepting the case of near boolean algebras about which no results came to our knowledge. Syuags and SOS-skeins. A mapping between two squags (SQS-skeins) is a homomorphism if and o n l y if it is a morphism of the underlying Steiner Triple Systems (Steiner Quadruple Systems).
Block ulgehras. q > 3 . Every homomorphism between two block algebras is a morphism between the underlying systems, but not vice versa. Nevertheless. if a morphism CC, from ( P I .R , ) t o ( P 2 .B,) and a co-ordinatization of (P?. B2) in a \ arietg of block algebras are given. then (PI.B,)can be co-ordinatized in such a way that q is a homomorphism. ( A little bit more can be said about automorphisms. see Section 7 . ) Sloops. Ekery injective sloop-homomorphism induces an injective morphism of the underlying triple systems. and vice versa. If q : ( S ;0. e ) + (7'; 0, e ) is not injective t h e n q ' ( { e } is ) a nontrivial "normal" subsloop of (S:.. e ) . Such subdoops are treated in the next paragraph. The effect on the underlying Triple System is: ( a ) The underlying subspace N of the normal subsloop cp ' ( { e } )disappears. ( b ) Two points which are not in N are identified by cp iff the block joining them intersects N. ( c ) Blocks n o t intersecting N are mapped onto blocks.
N e w wcror spaces. Here t h e situation is even more involved and no theory has lieen developed. Every 1- 1-homomorphism induces a 1- 1-rnorphism and every I - I-morphism is induced by some (but not every) co-ordinatization o f the s y t e m s . For homomorphisms which are not 1-1. the situation should be similar to that i n thc case of sloops.
Co-ordinatizing Steiner systems
1s
Fig. 3.
5. Congruence relations and normal subalgebras If cp : A + B is a mapping, the kernel ker cp is the equivalence relation on A defined by (a, b ) E ker cp
a
cp(a)= cp(b).
If q : (P, 13)-+ (Q, C ) is a morphism between Steiner Systems, ker cp is called a congruence relation on (P, B ) . If cp : ( A ,f;) + ( B ;F ) is a homomorphism between algebras, ker cp is called a congruence relation on ( A ; F ) . If 8 is a congruence relation, the classes of 8 are called congruence classes. The congruence class containing the element x is denoted by [ X I & Instead of (a, b) E 8 we also write a8b. If S is the underlying set, denote S/0 := {[x]8 I x E S}. Let ( A ; F ) be an algebra and 8 an equivalence relation on A. 8 is a congruence relation on ( A ;F ) iff Vfi E F V a , , . . , , a,,,b l , . . . , b,,, E A a,86,..
. .. a,,Ob,,,
3
f ( a , , . . . , a,,)8f(bl.. . . , bq).
For example, an equivalence relation 8 on A is a congruence on the groupoid (A;o) iff
vX,y, xi, yi E A
X w , YOY
3
x 0 yex'o y r .
If 8 is a congruence relation on the algebra ( A ; F),then .rr, :( A; F ) + (A/8; F )
is a homomorphism with ker T, = 8, where vfi€FVx,,...,x,EA
and vx
EA
.rr,(~) :=
XI^.
fi(Cx,le , . . . , C ~ l e ) : = C f i ( X l ., . . , x,)18
B. Canter. H . Wenier
I6
( A / O ;F) is the quotient algebra of ( A ; F ) modulo 8.
Varieties are closed under the formation of quotient algebras (i.e. homomorphic images): Quotient algebras of sloops are sloops, etc. Let ( P . 5 ) be a Steiner System of type (2, k ) and 8 be an equivalence relation on P. 8 is a congruence relation on ( P . B ) iff VS. v. y ' ,
yay'
Z G P
3
((x.
y ) n [ ~ ] ~ = p ( s ( ,xq. n [ ~ ] e = p ) .
Here (x. y ) denotes the subspace generated by x and y. If 8 is a congruence relation of a Steiner System (P. B ) of type (2. k ) . then 7 , :(
is
P. R
-+ (
P/8. B/B)
a morphism with ker nn = 8. where r e ( x ) = [ x ] O for all x E Y and R/8:={nn(h)I b E B . lne(b)\>l}.
( P / f l .B/O) i5 called the quotient system of ( P , B ) modulo 6. Now we list some properties which congruence relations may or may not have: i Pernnurubilify. If 6 and
I
4 are congruence relations, then 8 0 4=~ $08.
I
j
j Regularity. If two congruence relations have a common class then they are
j equal. ~
Uniforriiiry. All classes of a congruence have the same cardinality.
11 Coherence. ~
If a subalgebra (subspace) contains a congruence class, then it is the union of congruence classes.
In particular t h e property of permutability has many structural consequences. For example, if two congruence relations 6 and t,b permute then 8ot,b is also a congruence. In t h e case of varieties of algebras, permutability implies that the lattice of congruences is modular, and many further properties. Block
Near vector spaces (sloops)
algebras (squags) Permutability Lattice of congruences Regularity Uniformity Coherence
SQS-skeins
Steiner Systems t y p e (2, k ) yes
yes modular Yes yes Yes Each congruence class is a subalgebra ~
* For
Near boolean algebras
modular Yes Yes Yes The congruence class containing 0 is a subalgebra
distributive Yes Yes yes No congruence class is a subalgebra (except e = Q x Q )
modular Yes yes Yes Each congruence class is a subalgebra
* yes yes Yes Each congruence class is a subspace
~~~~~
Steiner Systems of type (2. k ) , k > 3. the intersection of two congruence relations necessarily a congruence relation.
IS
not
i i
Co-ordinalizing Steiner systems
17
A subalgebra is called normal if it is a congruence class. From the above table it follows that for the co-ordinatizing varieties the congruence relations are uniquely determined by the normal subalgebras. In other words: For squags, sloops, etc. the normal subalgebras play the same r81e as the normal subgroups for groups. We give some characterizations of normality for sloops, squags and SQS-skeins due to R.W. Quackenbush and M. Armanious. The first one can be derived from the more general result that a subloop N of a commutative loop ( L ;0, e ) is normal iff vx, y € L
xo(y"N)=(xoy)oN.
Note that whether or not a subsystem ( S , C)of (P, B ) corresponds to a normal subalgebra does not depend on the structure of C but only on B \ C. x = y (mod N ) abbreviates that (x, y ) is in the congruence corresponding to N.
(AS ;subsloop 0. e) is normal N of iff a sloop for all x. y. z E S
x o yN ~ J ( x o z ) o ( y o ~ ) EN.
i..\J
;;;yN)iff
A subsquag N of a squag
( S : is normal iff for all x, y, z E S and w E N (xow)oy~N+ ( ( x ~ z ) ~ w ) ~ (Ny ~ z ) ~ 0 )
x-y(modN)iff xoa
3a, b E N
=
y 0b
x=y(rnodN)iff 3 a E N q(a,x, y ) E N (iff q(x, Y. N ) c N ) .
As immediate consequences we obtain: Every subsloop of ( S ; 0, e ) of cardinality :IS- 11 is normal. If two disjoint subsquags of ( S ; . ) both have cardinality $IS], then both are normal. Every sub-SQS-skein of (Q; q ) with cardinality
is normal.
18
R . (;aitrrr, H . Wenier
One-element subalgebras of an algebra ( A ;F ) and the algebra A itself are always normal, they correspond to the trivial congruence relations id, and A x A. If an algebra has no other congruence relations it is called simple. From the uniformity of the congruence relations we can deduce that every co-ordinatking algebra of prime cardinality is simple. even more: If 11 is t h e cardinality of a finite nnnsimple sloop then there are admissible numbers s and t greater than 0 with t i = ( s + I ! ( r + 1 J . If t i is t h e cardinality of a finite nonsimple squag or SQS-skein then there arc admissible numbers s. r > 1 with t i = sr. Quackenbush proved that for every finite Steiner Triple System either the co-ordinati7ing sloop or t h e co-ordinatizing squag is simple. He has also shown that t h e onlv nonsimple finite planar sloop has ti elements and the only nonsimple finite planar squag has 9 elements. Moreo\.er. he proved that everv finite simple sloop with more than 2 elements a n d e\ery finite simple squag with more than 3 elements is functionally complete. kl. Mendelsohn has t h e result that t h e sloop o f the smallest non-derived finite triple system must be simple.
6. Subvarieties So far we were concerned with the variety o f all sloops. the variety of all squags. etc. By imposing further identities we can form subvarieties. E.g. wc might consider t h e variety of all squags satisfying the associative law x . o ( V o z ) = (s0 y 1 0 2 . and would easily find that this variety is t h e rriuial subvariety (because
can be deri\ ed from these identities). I t is not always that disappointing, there are nontrivial suhvarieties of the varieties under consideration. In fact. there are very many as the following theorem (due to Quackenbush) shows:
Theorem. The Ititrice of carieties of sloops (squags) contains as a cowr preserving sirhiartice the larrice of all subsers of a coutlrahle ser. I f M C take any algebra ( A :F ) i n one of o u r varieties we may consider the \ ariet! genertcfed by rhis algebra. i.e. the variety defined by all identities that hold in ( A: F ) . This is always a subvariety. in most cases ( e . g .when ( A :F ) is finite) it is a proper subvariety. Each of the co-ordinatizing varieties has a stnallest nontrivial subvariety. that means a subvariety which is contained in all other subvarieties. Each of thew whvarietie\ is generated by the smallest nontrivial algebra in t h e respective class:
Co -ordinatizing Sfeiner systems
19 ~
~~
The variety generated by the 2-element sloop is the variety of all boolean groups. It is characterized among all sloops by the associative law x o ( y 0 z ) = (x0y)oz. The underlying triple systems are the projective spaces over GF(2). The variety generated by the 3-element squag is the variety of all groupoids ( V; o ) , where V is a vector space over GF(3) and x y : = 2x + 2y. It is characterized among all squags by the medial (or entropic) law ( W O X ) ~ (yoz)= ( w ~ y ) ~ ( x ~ z ) . The underlying triple systems are the affine spaces over GF(3). 0
The variety generated by t h e q-element near vector space over GF(q) is the variety of all vector spaces over GF(q). It is characterized among all near vector spaces by the associative law. The underlying Steiner Systems are the projective spaces over GF(q). The variety generated by a q-element block algebra over GF(4) is the variety of all groupoids ( V ; 0) where V is a vector space over GF(4) and t h e operation 0 is defined by x o y = y + (x - y)a. It is characterized among all block algebras over the same algebra by the medial law (wox)o(yoz)= ( w o y ) o ( x o z ) . The underlying Steinet Systems are the affine spaces over GF(4). The variety generated by the 2-element near boolean algebra is the variety of all boolcan algebras. It is characterized among all near boolean algebras by the associative law (X A y ) z ~= x A ( y A z ) . The underlying triple systems are the projective spaces over GF(2). The variety generated by the 2-element SQS-skein is the variety of all algebras ( V ; 4). where V is a vector space over GF(2) and q ( x , y, z ) = x + y + z . It is characterized among all SQS-skeins by the identity 4 ( u , x, q(u, y, 2 ) ) = 4 ( x . Y, z). The underlying Steiner Quadruple System of such an SQS-skein has as blocks the affine planes of the vector space V.
For block algebras over proper nearfields the situation is not yet clarified. The next step would be to consider varieties which are generated by other co-ordinatizing algebras. The authors know of only one case where this has been done: R.W. Quackenbush has studied the variety generated by the 7-element squag.
20
R . Ganter, H . Werner
Some work has been done concerning block algebras over GF(q). For every q > 3 the two distributive identities
together imply t h e medial law
(Soublin) and thus force the underlying Steiner System to be an afine space. This is not true for q = 3: The underlying triple systems of the squags satisfying both distributive laws are exactly the afine Triple systems, i.e. triple systems in which every triangle generates an affine plane. The algebraic properties of these systems are well studied, see Klossek and Young. T. Kepka (private communication) was able to construct for each q 3 5 finite block algebras which satisfy one of t h e distributive laws but not the other. This is a rather surprising result and the underlying Steiner Systems should be of conibinatorial interest because of their (transitive) automorphism group: Every point is the (only) fixed point of some automorphism. In (91 block algebras over G F ( q " ) are studied which have t h e property that the operation * defined by
is medial (this can. o f course be formulated as an identity for t h e operation 0, these algebras thus form a subvariety).The underlying Steiner Systems are exactly those of type (2. q " ) whose blocks can be represented by subspaces of a vectorspace over GF(q). If * satisfies. in addition,
thc bariety obtained co-ordinatizes exactly the (2, q")-translation spaces (cf. Rarlotti & C'ofnian). I n [9] some applications to translation planes and spreads are mentioned.
7. Some remarks on automorpbisms One might hope that there is an algebraic description of t h e automorphism group of a Steiner System, at least if the system was constructed via the direct product of co-ordinatizing algebras. In fact one has:
Co-ordinnrizing Steiner systems
21
Lemma. Let -Y. be a variety of block algebras or the variety of SQS-skeins. If ( A ; F ) , (B; F ) E ?f are simple algebras which are not embeddable into each other, then Aut((A; F ) x ( B ;F))=Aut((A; F))xAut((B; F ) ) .
In the case of squags and SQS-skeins the algebraic automorphisms coincide. For example, E. Mendelsohn used the language of squags and SQS-skeins in his proof that every finite group is the automorphism group of some finite STS and some finite SQS. For other block algebras the situation is quite different: Suppose (P, €3) is a Steiner System of type (2, q), q >4, and let cp be an automorphism of (P, B ) not fixing all blocks. Then there is a block algebra co-ordinatizing (P, B) for which q is not an automorphism. In [9] some attempts were made to obtain an algebraic description of the automorphisms of (P, B) by introducing the concept of a semilinear mapping: A mapping cp : P+ Q between two block algebras over GF(q) is called semilinear if there is an integer n < q such that
Theorem. Let ( P ; " ) and (Q;.) be nontrivial block algebras over GF(q) and suppose that every desarguesian plane in the underlying systems is co-ordinatized medially. Then every automorphism of the Steiner System co-ordinatized by ( P ;0) x ( Q ; is semilinear. 0)
Corollary. Every automorphism of the desarguesian plane of order q is a semilinear self-map of (GF(q); 0 ) x (GF(q); 0).
8. Derived operations Quackenbush has observed that there is a very simple connection between near vector spaces and block algebras over GF(q): If (Q; +, 0) is a near vector space over GF(q), define a groupoid (Q;.) by x o y := y
+ (x - y ) - a
for some suitable element a. Then (Q;.) is a block algebra. As an immediate consequence one has a construction3 which produces from every S ( 2 , q + 1, u ) an S ( 2 , q, ( q - l ) u + 1). This is an algebraic version of Quackenbush's "Idempotent Reduct Theorem". For more details cf. Quackenbush's article in this book.
_71
B. Gamer, H . Werner
The tZ.q)-system constructed in this way contains a central point, i.e. a point with t h e property that every triangle containing this point generates a (desarguesian) affine plane. I t seems likely that from a block algebra with a central point (1 near vector space can be reconstructed. This ha5 been studied in the case q = 3, where the following holds:: 1-et ( P ; 0 ) be a squag and c E P be a central point. Define o n P a binary operation @ by
Then ( P ;@. c ) is a near vector space o v e r GF(3). This near vector space is usually called the internal loop of t h e triple system. The internal loop satisfies t h e moufang identity ( . r @ ( y @ z ) ) @ x= ( x @ y ) @ ( z @ x ) iff the corresponding squag is self-distributive. and is an elementary abelian 3-group iff t h e squag is medial. Young and Klossek have applied Bruck's general t h e o r y o f nilpotency for loops to t h e internal loops of affine triple systems. and obtained rather sophisticated results on the lattice of subvarieties o f these systems. Nothing similar is known for the other co-ordinatizations, except that from each co-ordinatizing algebra a loop structure can be derived: neutral element of the loop
loop multiplication x c Y
t*
s F P arhitrary (I
s F I' arbitran
\ + Y
[[x. 51'. [v. sy1'. u - I - ( I h . (1
'
whcre -
a"'.
(I \
c I' arhitrar!
I t is completely unknown if thcsc or similar structures lead to further coordinatizations o!' Steiner Systems. lt seems likelv that there are such. We close the paper with t w o theorems showing that there are n o straightforward generalizations o f near vector spaces and of block algebras:
Theorem (Quackenhush ). Let 'I' he ari equational class with an esseritial binury polyriomial. Suppose that I ' has the property that euery 0-generared algebra h a s exactly 1 -elernenr. every strictly 1-generated algebra has exactly i)i elernerirs and evrry srrictly 2-gerierated algebra has exactly ti elements. Then Pn is a prinie power U t l d I1 = ) ? I 2 .
Theorem. 1-rt ( A ; F ) and ( R ;F ) be nontricial algebras such that the set of r -gerirrared subalgebras of ( A; F ) :< ( R ; F ) forms a Steiner System S(r, k . LI) with r -< L L. Theri these algebras are block algebras or SQS-skeins.
Co-ordinafizingSrcinrr systems
23
9. Some open questions ( 1 ) R.W. Quackenbush remarked that it should be straightforward to dcfine and study near vector spaces over nearfields. Is it? (2) Describe the variety generated by t h e smallest nontrivial block algebra over a proper nearfield 9. Are the underlying Steiner Systems just "affine spaces over 2P" (e.g. in the sense of Andre)? (3) Give a combinatorial description of t h e "near boolean algebra product" of Steiner Triple Systems. ( 3 ) Do the congruence relations of a Steiner System in general form a lattice'?
( 5 ) Give a simpler characterization of normal sub-SQS-skeins. (6) Is the assumption about the subplanes in t h e theorem in Section 7 dispensable? (7) Find a common generalization o f the two theorems in Section 8 ( 8 ) Is there a generalization of the co-ordinatization via near boolean algebras'?
References hot all references listed here are quoted in the text. We have also included some which arc of historical interest or which may be useful for further research in this field. [ I J J . Andri.. Affine Geometrien uber FastkBrpcrn. Mitt. Math. Sem. Univ. Gie.;scn 1 1-1 (1975,
I-w. [ 21 1 l.-J. Arnold. Zur Algebraisierung allgemeiner aHincr und zugehariger projektivcr Strukturen mit Hilfe eines vektoriellen Kalkuls. I n : Arnold et. al., cds.. Beitrage 7ur Gconietri\chen Algebra. l3irkhauser Verlag ( 1977) 25-29. 131 A. Rarlotti and J . Cofman. Finite Sperner spaces constructed from projective and aRnc spaces. Ahh. Math. Sem. Hamburg 40 (1973) 731-241. 1-11 R.H. Bruck. A survey of binary systems (Springer Verlag. Hcidelbcrg. 1971). [5] R.H. Bruck. What is a loop? in: A.A. Albert, ed.. Studies in Modern Algebra (Prcntice Ilall. Englewood Cliffs, NJ. 1063) 59-09. [ 6 ] T. Evans. Universal-algebraic aspects of combinatorics. ['reprint ( 1977). [7] B. Gantcr. Comhinatorial designs and algebras. t o appear in Algebra Unibersalis. [ X I 6 . Ganter, Kombinatorische Algebra (1.ecturc Notes. Darmstadt, 107-1). 191 B. Canter and R. Mety. Kombinatorische Algebra: Koordinatisierung von Hlockplanen. in: Arnold e t al., eds.. Beitrage zur Geometrischen Algebra (Rirkhauser Verlag, Rasel, 1977) I 11-124. [ l o ] B. Ganter and H. Werner, Equational classes of Steiner Systems. Algebra IJniv. 5 (197.5) 125-140. 1111 6. Canter and H. Werner, Equational classes o f Steiner Systems [I. Proc. Conf. Algebraic Aspects of Combinatorics IJniv. of Toronto (1975) 283-285. [I21 D. Johnson, A (2, 11) combinatorial groupoid, Discrete Math. 19 (1977) 265-271. [13] S. Klossek, Kommutative Spiegelungsraume, Mitt. Math. Sem. LJniv. Giessen 117 (1975) [ 141 E. Mendelsohn. The smallest non-derived triple system is simple as a loop. Algebra Univ. X (1978) 2 5 6 2 5 9 . [ 151 N.S. Mendelsohn. Combinatorial designs as models of universal algebras. Recent Progress in Combinatorics. Proc. Third Waterloo Conf. on Comb. (1968) 123-132.
24
B. Ganter,
H. Werner
[ 161 J . M . Oshorn. Vector loops, Illinois I . Math. 5 (1961) 565-584. [ 171 N.K Pukhare\.. Construction of At-algebras. Siberian Math. J . 7 (1966) 577-579. I1 X I R.W. Ouackenhush. Alprhraic aspects of Steiner Quadruple Systems. Proc. Conf. Algebraic Aspects of Comhinatorics. LJniv. of Toronto 11975) 265-268. 1141 R.W. Ouackenbush. Linear spaces with prime power block sizes. Arch. Math. 18 (1977) 38 1-386. [?!I] R W Ouackenhush. Near boolean algebras I: Comhinatorial aspects. Discrete Math. 1 0 i1074) 30 1-3OX. [ Z l R.W. Ouackenhush. Near vector spaces o \ e r GF(q) and (t.q + 1. I)-HIHL>s. Linear Algehra and Appl. 1 0 ( 1 0 7 5 , 259-266. [??I R.W. Quackenhush. Varieties o f Steiner loops and Steiner quasigroups. Canad. J . Math. 28 11976) 11x7-119s. (2.31 S.K. Stein. Homogeneous quasigroups. Pacific J. Math. I4 ( IO64I I(t91-I 102. [?-I] 1.. Szamktgowici. Sulla generalizzazione del concetto delle algehre A:. Atti Accad. Naz. Lincei Rend. Cl. Sci. Pis. Mat. Natur. ( 8 ) 78 (1965) 810-813. j 25 1 1. S 7 a m k c ~ o w i c t 0. 1 1 Steiner manifolds. Colloq. Math. 20 (1969) 45-5 I 12hj tf Werner. A unique factorization iheorein for Steincr Triple Systems. Preprint. (271 E. Witt.
Annals of Discrete Mathematics 7 (1980) 25-35 @ North-Holland Publishing Company
ALGEBRAIC SPECULATIONS ABOUT STEINER SYSTEMS Robert W. QUACKENBUSH* Departmenf of Mathematics, Uniuersity of Manitoba, Winnipeg, Manitoba, Canada, R 3 T 2 N 2 .
1. Introduction In recent years, techniques of universal algebra have been used extensively in the study of combinatorial design problems. Are these tools essentially new or are they merely algebraic reformulations of known combinatorial tools? The answer is: some of each. But even the recasting of a combinatorial technique in algebraic language can advance combinatorial design theory if it suggests other algebraic techniques with no known combinatorial counterparts, or if it suggests an obvious and useful generalization (the bane of combinatorial design theory has been the plethora of seemingly isolated techniques). A case in point is R.M. Wilson’s solution [ 141 to the existence conjecture for PBD’s. In large measure, his success was due to his ability to systematize known constructions; his first published proof of the conjecture has an algebraic as well as a combinatorial flavor. In this article I wanf to speculate about how algebraic ideas may be useful in telling us more about combinatorial designs in general and Steiner systems in particular. But first I want to mention two examples of algebraic techniques which provided the initial proofs of significant results in combinatorial design theory and one case where more algebraic knowledge would have prevented an embarrassing mistake. The reader should be familiar with the material in the article by B. Ganter and H. Werner in this volume. The following notational conventions are used in this article:
S ( n ) is the class of all Steiner systems with block size n. S(n, m ) is the class of all m-element members of S(n). R ( n ) and R ( m , n) are the analogous classes of resolvable Steiner systems. P(n) and P ( m , n) are the analogous classes of pointed Steiner systems (see Section 3 for their definitions). S p ( K ) is the set of cardinalities of members of the class K. The letter q is used to denote a prime power.
* The author’s research
was supported by a grant from the NSERC of Canada. 25
K.W. Quackenbush
26
2. Three examples
O u r first example is Bruck’s incorrect direct product construction for Room squares (R.H. Bruck [l]). A Room algebra, ( R ; *), is a pair of idempotent commutative quasigroups, ( R: and ( R ; *), satisfying: 0 ,
0 )
[s 0 y
= 14
0
c and x
[ x = u and y = c]
*
y
=
or [ x
u
* c] implies
=c
and y
=
ul.
Bruck’s construction is equivalent to the assertion that the direct product of two Room algebras is again a Room algebra. R.C. Mullin and E. Nemeth [7] showed that this is not the case. The difficulty is that in general, direct products d o not preserve sentences of t h e form ( l j in which the conclusion of an implication is a disjunction of positive sentences. We are all familiar with this phenomenon with fields: the direct product of two fields is not a field since fields satisfy x y = 0 implies x = 0 or y = 0; hence in the direct product we have (0,O) = (1.0)(0, 1 ) while ( I , 0 ) # ( 0 . 0 )# (0. 1). Indeed, one easily sees that the direct product of non-trivial Room algebras is never a Room algebra. O u r second example is E. Mendelsohn’s result that every finite group is isomorphic to the automorphism group of some finite Steiner triple system; see E. Mendelsohn [6]. The proof is a very delicate interweaving of graph theory with the combinatorial structure of Steiner triple systems and the algebraic structure of Steiner loops. Mendelsohn first represents any given finite group, G, as the automorphism group of a 4-ary relation, R, on a finite set, X . Next he chooses a 16-element Steiner loop. S, which has no proper automorphisms and has no 8-element subloop. Then he considers B ( X ) , the boolean group with basis X (recall that boolean groups are Steiner loops; t h e corresponding triple systems are the projective geometries over GF(2)). Whenever ( x , , x 2 , x 3 , x,) E R he replaces the subloop generated by {x,, . . . , x,} with a copy of S. He then proves that the algebra so obtained is a Steiner loop whose automorphism group is isomorphic to G. While the proof can be presented without any reference to loops. the ideas and insights are clearly of an algebraic nature. O u r last example is R. Canter’s theorem [3] that any finite partial linear space can be completed to a finite linear space. The first step in this direction is C. ‘Treash’s proof [ 131 that any finite partial Steiner triple system can be finitely completed. Canter [2] then generalized this to show that every finite partial S ( q + 1 J can be finitely completed; his method was a projective geometric generalization of Treash’s. Then the author [9] showed that every finite partial S ( q ) can be finitely completed, thus breaking the mold of projective geometry. From this Canter [3] was able to devise a general method for finitely conipleting finite partial linear spaces. But the interesting point is that the author’s proof is an algebraic translation of the S ( q + 1) case to the S ( q ) case. First convert the partial S(4) to a partial S ( q + 1 ) by adding a new point to each block. Now complete the
Algebraic speculations abour Steiner sysrems
27
partial S ( q + 1) to, say, r elements. Next use the algebraic translation, the Idempotent Reduction Theorem (which will be discussed further in the next section), to obtain an S ( q , r(q - 1)+ 1);this can be done so as to retain the original blocks of the partial S(q), thus finitely completing it.
3. The idempotent reduction theorem We will state and prove this theorem in terms of Steiner systems. For this we need the concept of a pointed Steiner sysrem: ( S , B)ES ( n , u ) is pointed if there is a point p E S such that every two distinct blocks through p generate an affine plane. Let P(n, u ) denote the class of all pointed Steiner systems of size u with block size n.
Idempotent reduction theorem [9]. There is a natural correspondence befween S ( q + 1, u ) and P(q, (q - l ) u + 1).
Proof. Let (S, B ) ES ( q + 1, u ) with S = { p , , . . . , p , } and define S‘ = {pi,iI 1 c j s q - I , piE S}U (0). Define B’ as follows: for any block b = { p , , . . . , p,,,} in B, define an affine plane on
{p,,,I I S j S q - 1 , I s i S q + l } U { O }
so that for 1 s i s q + 1, {O}U{pi,i I 1S j S q - 1) is a block. Place all blocks of this affine plane in B’ and repeat this for each block of B. It is straightforward to check that (S’, B’)EP(q, (q - l ) u + 1) with 0 as the distinguished element. Conversely, if (S’, B’) E P(q, (q - l ) u + 1) with 0 as distinguished element, let P be the set of all affine planes containing 0. For p E P we consider p to consist of all affine lines in p through 0. Then one readily checks that (B’, P)ES ( q + 1, u ) . One possible application of this theorem is to the construction of S(6, u ) for small u, a notoriously difficult problem. The theorem tells us that an S ( 6 , u ) is equivalent to a P(5,4u + 1). Since S(5, u)’s are relatively easy to construct, it may turn out to be easier to construct a P(5,4u + 1 ) and to convert it to an S ( 6 , v ) than to construct the S ( 6 , u ) directly. Another use of the theorem could be made by showing that P(q) is closed under some recursive construction and then translating to S ( q + 1). For instance, it is trivial that P(q) is closed under direct products. Translating to S ( q + l),this tells us that u , , u, E S ( q + 1) implies (q - l ) u , u, + u1+ u, E S ( q + 1) which on the face of it appears to be a non-trivial construction. Of course, P(q, u)’s are special, but they are special in an interesting way:
Theorem. Each P(q, u ) is resoluable.
'X
R. W. Quackenbush
Roof. Let ( S , €2)
E P(q, u ) with distinguished element 0. Let a , , a2 E S and let (0, a , } € b,; we must describe the block 6 2 containing a , parallel to b , . But this is just the block containing a, in the affine plane generated by (0,a , , a,}. It is straightforward that this procedure gives a resolution of B into parallel classes.
What does this tell us about R ( q ) ? Of course, P ( q ) c R ( q ) cS ( q ) with each containment being proper. We know that u E S ( q + 1) implies u = 1 mod ( 4 ) and u ( u - 1 ) = 0 mod ( q 2+ q ) and that these conditions are asymptotically sufficient. Also, the number of residue classes mod ( q 2 +q ) which satisfy these two conditions is 2' where r is the number of distinct primes dividing q + 1; this number is less that q + 1. Thus when we translate to P ( q ) ,not all residue classes mod ( q 3 - q ) which are multiples of q occur. But for R ( q ) we know that the condition c = q mod (4'- q ) is asymptotically sufficient, so each residue class mod ( q 3- q ) which is a multiple of q occurs in R ( q ) . Hence Sp ( P ( q ) c ) Sp ( R ( q ) )c Sp ( S ( q ) ) with each containment being proper. For q = 3 we know that u = 1, 4 mod (12) is sufficient for S(4, u ) # $4. Hence c = 3 , 9 mod (24) is sufficient for P ( 3 , u ) # $3, We also know that u = 3 mod ( 6 ) (equivalently, u = 3 , 9 , 15,21 mod (24)) is sufficient for R ( 3 , u ) # $3. Thus when H. Hanani [ 5 ]showed that u = 1, 4 mod (12) was sufficient for S(4. u ) # fl, he solved half of Kirkman's Schoolgirl Problem. It would be nice to find some recursive constructions on R ( 3 ) (and more generally R ( q ) ) to generate designs with ) Sp ( P ( 3 ) ) . For instance, if u i E P ( q ) , uz E R ( q ) , then cardinalities in Sp ( R ( 3 ) L:,U~E R ( q ) . However, it seems difficult to find natural constructions on P ( q ) which yield members in R ( q ) but not in P ( q ) .This is analogous to the difficulty of finding constructions on P(6, u)'s with u = 1,6 mod (30) which yield a P ( 6 , u) with D = 16 or 21 mod (30). 4. Projective hyperplanes, a n e hyperplanes and maximal homomorphic images A proper subdesign (S', B')=( S , B ) E S ( q + 1) is a projective hyperplane if for every b E B. b n S'# $4. A proper subdesign ( S ' , B') c ( S , B ) E S ( q ) is an affine hyperplane i f for every p E S - S', (S,, BP)is a subdesign such that every b E B with exactly one point in S' has a point in S,, (here B,,= { b I P E b, b n S ' = $4) and S,, = { a I a E b E B,}). We next remind the reader that S ( q + 1) corresponds to the variety of all algebras such that every 2-generated subalgebra is a vector space over GF(q). This variety, V2(q), has a unique minimal non-trivial subvariety, namely "Ir(q),the Lector spaces over GF(q). Of course, V ( q ) is generated by V(q), the 1dimensional vector space over GF(q).See the article by B. Ganter and H. Werner i n this volumn. If q 2 3 , then we can have non-isomorphic A, A' in V , ( q ) corresponding to the same ( S , €3). For q = 2 however, if A and A' both correspond to (S. B), then A and A' are isomorphic. This distinction will become important later in this section.
Algebraic speculations about Sfeiner sysfems
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Theorem. Let (S’, B‘) be a subdesign of ( S , B)E S(q + 1); (S’, B’)is a projective hyperplane iff there is an A E V2(q)corresponding to (S, B) and a homomorphism, h, from A onto V(q) such that (S’, B’)corresponds to h-’(O).
Proof. Let h : A + V ( q ) be onto and let (S‘,B‘) correspond to h-’(0); clearly (S’,B’)is proper. Let b E B ;thus b corresponds to a 2-dimensional subalgebra of A, A,. Hence h is not 1-1 on A, and so some 1-dimensional subalgebra of A, is mapped to 0 by h. This means that b nS’#F, and so (S’, B’)is a projective hyperplane. Conversely, let (S’, B’)be a projective hyperplane; we must define A and h : A+ V ( q ) so that h-’(O) corresponds to (S’, B’).As above, we let S = {a, I i E I}, V ( a , )= (0, ai, a:, . . . , a:-’} where V ( a i )is isomorphic to V(q)with 0 as zero; let A = U { V ( a i ) Ii E I } . For u , E S ’ , define h(V(a,))=O.For a i E S - S ’ , define h : V ( a , )+ V(q)to be an isomorphism. Since (S’,B‘) is proper, h maps A onto V ( q ) . For b E B, let V ( b )= U{V(a,)1 ai E b}. Since the operations of ‘V,(q) are at most binary, h will be a homomorphism provided it is a homomorphism on each V ( b ) . If b E B’,then h(V(b))= 0, so n o matter how we define V(b)to be isomorphic to V(q)2,h will be a homomorphism. Let {ao, a,, . . . , aq}= b E B - B’. Since (S’,B’)is a projective hyperplane, there is a unique element of b, say a,, contained in S’. Thus h( V(a,)) = 0 and h( V ( a , ) )= V(q)for 1S i s q . In this case we can also make V ( b ) isomorphic to V(q),such that h is a homomorphism on V ( b ) .Thus A E V2(q)corresponds to ( S , B) and h is a homorphism from A onto V ( q ) such that h-’(O) corresponds to (S’,B‘). Thus we see that the projective hyperplanes of (S, B)are just the subdesigns of (S, B)corresponding to “kernels” of homomorphisms of algebras in Sr,(q) corresponding to (S, B) onto V(q). Corollary. Let q = 2 (i.e., we are dealing with Steiner loops). Let ( S , B ) E S ( ~ ) correspond to A E ‘V2(2). Then there is a 1-1 correspondence between the projective hyperplanes of ( S , B ) and the homomorphisms of A onto V(2).
Proof. This follows from the fact that there is a unique A E V2(2)corresponding to (S, B). Problem. Is it true that for every ( S , B)E S(q + l),there is an A E ‘V2(q) such that the projective hyperplanes of ( S , B ) are in 1-1 correspondence with the homomorphisms of A onto V(q)? For A E V2(2),we let A, be the maximal quotient (homomorphic image) of A in ‘V(2); i.e., if A’ is any quotient of A in ‘V(2), then A’ is a quotient of A,. Such an A, always exists and is unique. Since A, E ‘V(2), it is a vector space and in some sense is the largest vector space which can be modeled in A. Thus, for instance, the dimension of A is the dimension of A,; the sublattice of subspaces of (S, B) generated by the projective hyperplanes is the projective geometry corresponding to A,; in particular, (S, B) is a projective space (i.e., A, = A ) iff
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R.W. Ouackenbush
the intersection of all projective hyperplanes of ( S , B) is empty. The reader is referred to L. Teirlinck [12] where these and other variations are proved in the more general context of 2-coverings (where two points may lie in more than one block and where all blocks need not have t h e same size). A completely analogous situation occurs for affine hyperplanes. Recall that S ( q ) corresponds to dB?(q), the variety of all algebras such that every non-trivial 2-generated subalgebra is isomorphic to A(q), the 1-dimensional affine algebra over GF(q). Let d(q)be the variety of all affine spaces over GF(q) (vector spaces over GF(q) where the only operation is a x +(1 - a ) y ) ; d(q)is generated by A(q). Hence d ( 4 ) is the unique minimal non-trivial subvariety of d,(q).For qz-4,we again may have two non-isomorphic algebras in 7rz(q) corresponding to the same ( S , B ) E S(q). For q = 3 , the algebra corresponding to ( S , B ) is unique.
Theorem. Let ( A ' ,B') be a subdesign of ( A ,B ) ES ( q ) ; ( A ' ,B') is an amne hyperplane i f l there is an A E d,(q)corresponding lo ( A ,B ) and an a E A(q) and a homomorphism, h, from A onto A ( q ) such that A ' = h -'(a).
CoroUary. Let q = 3 (i.e., we are dealing with Steiner quasigroups). Let ( A ,B ) E S ( 3 ) correspond to A E d 2 ( 3 ) and let a E A. Then there is a 1-1 correspondence between the affine hyperplanes of ( A ,B )containing a and the homomorphisnis of A onto A(3).
The proofs are similar to the projective case and are omitted. Maximal quotients in d(3)and affine hyperplanes play an analogous role to maximal quotients in V ( 2 ) and projective hyperplanes. Also, for q > 4 we have an analogous problem.
Problem. Is it true that for every ( A ,B ) ES ( q ) , there is an A E sP2(q)corresponding to (A, B) such that for every a E A there is a 1-1 correspondence between the afine hyperplanes containing a and the homomorphisms of A onto A(q)'?
5. Almost all Steiner systems What percentage of Steiner triple systems have non-trivial automorphism groups'? To be more precise, let T ( n )be the number of (isomorphism classes of) triple systems of size s n and N ( n ) the number with non-trivial automorphism groups: what is the behaviour of N ( n ) / T ( n )as n - x ? Since every finite group is t h e automorphism group of a finite triple system it seems reasonable to expect N ( n ) / T ( n ) to tend to 1. Yet almost surely N ( n ) / T ( n )tends to 0. In fact, we conjecture that almost all triple systems have the following two properties: ( 1 ) (S. B )is planar; i.e., it is generated by every triangle (three points not in a block).
Algebraic speculations about Steiner systems
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(2) ( S , B) is rigid; i.e., it has a trivial automorphism group. Evidence for this conjecture comes from a theorem of V.L. Murskii [8] about finite groupoids (algebras whose only operation is binary). Murskii proved that almost all finite groupoids have these three properties: (i) they are simple, (ii) their proper subalgebras are all trivial, (iii) they are rigid. We can consider triple systems as either idempotent quasigroups or loops; in either case they are groupoids. Hence we can speculate about the extent to which triple systems inherit these properties of groupoids. Since almost no groupoids are quasigroups, we can make no direct application of Murskii's result. Indeed, neither Steiner quasigroups or Steiner loops satisfy (ii) since every triple (together with 0 in the case of Steiner loops) forms a subalgebra. However, these are the only non-trivial proper subdesigns that a triple system must have; hence we conjecture (1). What about simplicity? Since we do not associate homomorphisms with triple systems and simplicity is concerned with homomorphisms, we can ignore condition (i). But perhaps triple systems which are simple as Steiner loops or Steiner quasigroups have certain properties which we should consider. Fortunately, a planar STS is simple both as a Steiner quasigroup and as a Steiner loop [lo] (with one exception in each case). Thus (1) and (2) are the analogues for triple systems of (i)-(iii). Note that (1) and (2) make no reference to block size; hence we shall extend the conjecture to S(n): Conjecture. Almost all members of S(n) are planar and rigid.
6. Derived triple systems
In another chapter of this book, K. Phelps has given a survey of derived triple systems which we assume the reader to have read. In this section we will speculate about how one could prove that every triple system is derived. The reader will have noticed that t h e general results of Phelps' article are recursive in nature (e.g.. the direct product of derived triple systems in again derived). However, it is clear that a proof that every triple system is derived is unlikely to be recursive in nature. A direct construction seems needed, but it is totally unclear how to make such a construction. We will describe a model-theoretic approach which may facilitate thinking along the lines of a direct construction. The first step is to prove that every infinite triple system is derived. The block designer, accustomed to considering only finite designs, may consider this result uninteresting or irrelevant. But this construction has two useful features; it shows that there is some hope for a direct construction, and it enables us to prove that
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R . W. Quackenbush
the class of derived triple systems is axiomatic (i.e., it can be described by a set of first order sentences; what we would like to show is that this set can be taken to be just the sentences defining all triple systems).
Theorem. Every infinite triple system is derived. Proof. Let (S, T ) be an infinite triple system with O & S. Let S’= S U{O} and Q’= {I U{O} 1 I E T). Let T‘ be the set of triangles of S (i.e., 3-element subsets not in T); well-order T’ so that T’ = { f a .1 a ’ < a }where a = 19 .1. We assume inductively that for a”
Theorem. A class of structures i s axiomatic iff it is closed under prime products and elementary equivalence. Theorem. The class of derived triple systems is axiomatic.
Proof. Let ( S , T) be a prime product of the derived triple systems (Sz,T,),,,. If ( S , r ) is infinite, then it is derived; otherwise it is finite and hence isomorphic to some (Sl, T,) and so derived. Let ( S ’ , T’)be elementarily equivalent to the derived triple system ( S , T ) . If ( S ’ , T’) is infinite, then it is derived; otherwise it is isomorphic to ( S , T> and so derived.
Algebraic speculations about Sreiner systems
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We now describe a useless axiom system for derived triple systems. For every finite triple system, (S, T), there is a first order sentence, w(S, T), which states that for any triple system (S', T'), that (S', T') is not isomorphic to (S, T). Hence the axiom system is (i)-(iv) and {w(S, 7') I (S, 7') is a finite triple system which is not derived}. However, it is conceivable that someone can give an explicit set of sentences characterizing derived triple systems. One would then try to show that these are all consequences of the axioms for all triple systems. Alternatively, one would try to construct a triple system not satisfying all the axioms for derived triple systems. In either case, an explicit axiom system would likely shed much light on the current murky situation. A class of structures is elementary provided it is defined by one first order sentence; note that this is equivalent to being defined by a finite set of first order sentences since this set is logically equivalent to its conjunction.
Corollary. The class of deriued triple systems is elementary iff there are only finitely many triple systems which are not derived. Proof. If there are only finitely many triple systems which are not derived, then the useless axiom system described above is finite. If there are infinitely many triple systems which are not derived, then they have unbounded finite size. From this we can conclude that a prime product of all these non-derived systems would be infinite and hence derived. But if derived triple systems were elementary, then every prime product of non-derived triple systems would be non-derived. Thus if there are infinitely many non-derived triple systems, then derived triple systems are not elementary. It is conceivable that someone can find a construction which shows that for some n, every triple system of order at least n is derived. Alternatively, someone might be able to provide an explicit finite axiom system for derived triple systems. This would show that there is an n such that every triple system of order at least n is derived. As mentioned by Phelps in his article, it is well known that the direct product of derived Steiner quasigroups is derived. We next show that the direct product of derived Steiner loops is derived; from an algebraic viewpoint, this is a triviality. Note that for triple systems of order u1 and u2, the triple system obtained from the loop direct product has u,u2 + u1 + u2 elements. Recall that quadruple systems correspond to the variety, V, of Steiner 3-skeins (algebras of the form ( A ; m ) where m(x, x, y ) = m(x, y , x ) = m(y, x, x ) = y and for distinct a, b, c E A, m(a, b, c ) is the other element in the unique quadruple containing a, b and c ) . We next form V*, the free constant extension of V; V* = { ( A ;m, a ) 1 ( A ;m ) E V and
a E A}
R. W. Quackenbush
31
(i.e.. we label in each possible way one element in each Steiner 3-skein). It is easy to show that V* is a variety. Next we form a class of reducts of V*: Vd = { ( A ;m ( x , a, z ) ) I ( A ; m ( x , y,
2).
a > €V*).
One easily checks that V" is a subclass of the class of Steiner loops, and as such is just the class of Steiner loops which are derived triple systems. It is a universal algebraic triviality that V d is closed under direct products; hence derived triple systems are closed under loop products. Consider the algebra Z(2) = ({O, 1);m ) where m(x, y, z ) = x + y + z mod 2. It is well known and easy to prove that the variety V of the last paragraph is the variety of all algebras satisfying all 3-variable identities true in Z(2). The variety of Steiner loops is the variety satisfying all 2-variable identities true in the 2-element group ({O, 1); +). If it is possible to prove that all triple systems are derived, then given any Steiner loop ( A ;+), one would expect to be able to define a Steiner 3-skein ( A ; m ) in terms of +; moreover, one would expect this definition to be uniform; i x . , that rn be given by some polynomial in +. There are two obvious candidates: ( x + y ) + z and x + ( y + z ) . Neither of these work in general. One reason for this is that m is completely symmetric; hence if either worked, then both would work and equal m. Hence + would be associative. But if + is associative. then ( A ;+) is a boolean group. All boolean groups are derived; in fact. for + associative, ( A ;+) is derived from ( A ;x + y + z). This is a good indication that there is n o uniform proof. This has lead the author to conjecture that not every triple system is derived.
References [ 11 R.H. Bruck. What is a loop, Studies in Modern Algebra, Math. Assoc. of Amer. (1963) 59-99. [2] B. Ganter. Endliche Vervollstandigung endlicher partieller Steinerscher Systeme, Arch. Math. 22 (1971) 32X 332. [3] B. Ganter, Partial pairwise balanced designs, Colloquio Internazionale sulle Teorie Combinatoire (Roma, 1973). Tom0 11. Atti dei Convegni Lincei, No. 17. Accad. Naz, Lincei, Rome (1976) 377-380.
[4]G. Gratzer. Universal Algebra, 2nd ed. (Springer-Verlag, New York, 1979). [S] H . Hanani, The existence and construction of balanced incomplete block designs, Ann. Math. Stat. 32 (1961) 361-386. [h] E. Mendelsohn. O n the groups of automorphisms of Steiner triple and quadruple systems, J. Combinatorial Theory ( A ) 25 (1978) 97-104. 171 R.C. Mullin and E. Nemeth. A counter-example to a multiplicative construction of Room squares, J. Combinatorial Theory 7 (1969) 264-265. [ X I V.L.. Murskii, The existence of a finite basis of identities, and other properties of "almost all" finite algebras (Russian), Prob. Kibernet. 30 (1975) 43-56. [9] R. Quackenbush, Near-vector spaces over GF(q) and (c, q + 1)-BIBDs, Linear Alg. and Appl. 10 (1975) 259-266. [ 101 R . Quackenbush, Varieties of Steiner loops and Steiner quasigroops, Canad. J . Math. 2 8 (1976) 1187-1198.
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[ l I ] W. Taylor. The fine spectrum of a variety, Alg. Univ. 5 (1975) 263-303. [12] L. Teirlinck, On projective and afine hyperplanes, preprint. [13] C. Treash, The completion of finite incomplete Steiner triple systems with applications to loop theory. J . Combinatorial Theory 10 (1971) 259-265. [14] R.M. Wilson, An existence theory for pairwise balanced designs I, Composition theorems and morphisms, J. Combinatorial Theory 13 (1972) 220-345.
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Annals of Discrete Mathematics 7 (1980) 37-39 @ North-Holland Publishing Company
ALMOST ALL STEINER TRIPLE SYSTEMS ARE ASYMMETRIC Laszl6 BABAI Department of Algebra, Eiiroiis University, Budapest, Hungary and University of Montreal, Canada
Employing recent progress on the van der Waerden Permanent Conjecture, we prove the well-known conjecture that almost all Steiner triple systems are asymmetric (automorphism free).
Introduction Let n denote a (large) positive integer congruent to 1 or 3 mod 6. Let S ( n ) and S ” ( n ) denote the number of labelled and unlabelled Steiner triple systems (STS) on n vertices, respectively. We call an STS asymmetric if it has no non-identity automorphisms. Let A(n) and Ao(n)denote the number of asymmetric labelled and unlabelled STS’s, respectively. Asymmetric STS’s of each admissible order n 2 15 have been constructed by Lindner and Rosa [3]. They and other authors (cf. [4],for instance) conjectured that almost all STS’s are asymmetric.
Theorem 1. Almost all Steiner triple systems are asymmetric, i.e. A(n)-S(n) and A”(n ) So( n ) . In particular, we have S o ( n ) S( n ) / n! (- indicates asymptotic equality while n -+ m.)
-
-
In fact, we shall prove that the probability that a random (labelled or unlabelled) STS is asymmetric is extremely high; it exceeds 1- n-n2(1’16co(1)). The proof employs V.E. Aleksejev’s [l] and R.M. Wilson’s [5] lower bound on S ( n ) . They prove S ( n ) > n-n2(1’12+d1)). This estimate is sufficient to prove that the only prime divisors of the order of the automorphism group of almost every STS are 2 and 3. However, we need a stronger estimate to rule out these two primes. It is implicit in [l] and explicitely stated and proved in R.M. Wilson’s paper [5] that log S ( n ) n2 log n/6 assuming the van der Waerden Permanent Conjecture (v.d.W.P.C.). This conjecture asserts that the permanent of a k x k doubly stochastic matrix is Pk 3 k ! / k k .In fact, the weaker inequality p k >ePkis sufficient for Wilson’s logarithmic asymptotics for S ( n). Recently, Shmuel Friedland proved P k 2e-‘ [2], a major breakthrough towards the v.d.W.P.C. Hence now we have
-
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38
Theorem 2. (R.M. Wilson-S. Friedland).
holds trivially.)
(Note that the upper bound S ( n ) <
It will be fairly easy to derive Theorem 1 from Theorem 2.
Proof of Theorem 1. Let B ( n )= S ( n ) - A ( n ) and B o ( n )= S " ( n ) - A o ( n ) . Let us fix an n-element set V, the vertex set of the S(n ) labelled STS's. Let ZI denote the set of those permutations of V having prime order. For 7~ E 17,let B(r,n) denote the number of those STS's on V admitting r as an automorphism. Clearly, B ( r l ) S L E 1B(7T. , n). Let r E IZ. Let f denote the number of fixed points of r . If r is an automorphism of an STS then the fixed points form a (proper) subsystem, hence f s (n - 1)/2 (otherwise B ( z , n)=O). Now B(r,n ) can be estimated from above as follows. There are S(f, ways to select the subsystem on the set of fixed points. (We set S ( O ) = S( 1) = 1). We proceed by assigning third points to the remaining (;)-(:) pairs. These pairs are divided into p =lrl classes (the orbits of 7 ~ ) .Once we selected a triple, p - 1 others are forced to occur unless p = 3 and the triple selected is one of the orbits of 7r. Hence, we have a choice of assigning the third point to a pair only
times if p # 3 and at most ri-f
1 9
_-+ -
3
((") 2
-
(2')
- ( n-
n)
times if p = 3. In any case, we obtain
<
(1)
log n/3 +(n2-f'
+ n - f ) log n/6p
< n 2 log n( 1+ 3/p)/24 (using f ~ (- n1)/2). We infer
= ((5/48)+0(
l ) ) n 2 log n.
Almost all Steiner t i p l e systems are asymmemc
39
Combining this inequality with Theorem 2 and the trivial inequality So(n)> S ( n ) / n ! we obtain max ( B ( n ) / S ( n )B , o ( n ) / S o ( n )< ) n ! B ( n ) / S ( n )< n-(1116+d1))nz This proves A ( n ) - S ( n ) and A o ( n ) - S o ( n ) . As obviously A"(n)=A(n)/n!, the last assertion of Theorem 1 follows as well. 0
Remark. We also observe that the probability that the automorphism group of a random STS has n o automorphism of prime order >q is less than exp (-n'log
n(l/6-(1+3/q)/24+0(1))= n-(1-11q-cd1))n2'8
References [l] V.E. Aleksejev, 0 M e sistem trojek Stejnera, Mat. Zametki 15 (1974) 767-774. (English translation: On the number of Steiner triple systems, Math. Notes 15 (1974) 461-464.) [2] S. Friedland, A lower bound for the permanent of a doubly stochastic matrix, Ann. Math. 110 (1979) 167-176. [3] C.C. Lindner and A. Rosa, On the existence of automorphism free Steiner triple systems, J. Algebra 34 (1975) 430-443. [4] R.W. Quackenbush, Algebraic speculations about Steiner systems. [S] R.M. Wilson, Nonisomorphic Steiner triple systems, Math. Z. 135 (1974) 303-313.
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PART I1
STEINER SYSTEMS WITH HIGHER VALUE OF
t
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Annals of Discrete Mathematics 7 (1980) 43-63 @ North-Holland Publishing Company
EXTREMAL RESULTS AND CONFIGURATION THEOREMS FOR STEINER SYSTEMS Peter J. CAMERON Menon College, Oxford OX1 4JD. England
1. Introduction In the study of Steiner systems, most effort has been directed towards questions of existence and, to a lesser extent, enumeration. The point of view of this article is different, though related. The theme is structural properties of Steiner systems, and especially the characterization of special systems by structural properties. A prototype of this is the characterization of classical projective planes by the “theorems” of Desargues and Pappus. Two particular systems, those with parameters S(5,8,24) and S ( 5 , 6 , 12), will be very important in this paper. They and their first contractions were until recently the only known S(f, k, u ) with r >3; and they have a good claim to be regarded as “classical”. They have many striking properties. One of the best ways to illustrate these properties is through different constructions; and Section 2 is devoted to sketching several of the known constructions of these systems. Throughout the sequel, they reappear as the subject matter of various characterizations. They also crop up in other areas of mathematics, such as the theory of error-correcting codes [31], and the construction of finite simple groups [ 121. The next two sections discuss inequalities connecting the parameters of a Steiner system, and the determining of systems which attain the bounds. In Section 3, two simple inequalities are proved. In the first, the systems attaining the bound are projective planes and their extensions; the latter are completely known with the possible exception of one parameter set. In the second, both S(5,8,24) and S(5,6,12) meet the bound, but the situation is less well understood. The work of Gross on the numbers of blocks having prescribed intersection with a given block is described in Section 4. The material of the next three sections concerns some very geometric classes of Steiner systems: projective and affine spaces, projective, affine and inversive planes, and unitals. In the spirit of this paper, several classical characterizations of these systems are given. Another topic is the embeddability of Steiner systems in projective spaces or planes. In Section 8, we describe a simple property, the symmetric difference property, 43
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P.J. Cameron
shared by S(5,8,24) and S(5,6, 12); it seems that it may characterize these systems along with the binary affine spaces. The last section discusses possible generalizations and further directions for research: subspaces, matroid designs, coordinatizations, association schemes, and automorphism groups.
2. Constructions of S(5,8,24) and S66,U )
The existence and uniqueness of the remarkable Steiner systems S(5,8,24) and S(5.6, 12) has been known for nearly fifty years, while their automorphism groups were discovered a century ago. Since these systems figure prominently in the rest of this paper. This section is devoted to outlining a number of constructions of them. The constructions also illustrate various properties of the systems, and many of them can easily be modified to give uniqueness proofs. The groups now known as M,, and M,,, were discovered by Mathieu [35], who simply wrote down permutations which generate the groups. A more satisfying construction was given by Witt [49]. He observed that the stabilizer of three points in M,, is PSL(3,4), the unique simple normal subgroup of Aut S(2,5,21) (the projective plane of order 4). Witt proved a sufficient condition for a doubly transitive group to have a transitive extension, and showed that this condition could be applied three times to PSL(3,4), yielding MZ4. Witt also proved a sufficient condition for a 1-fold transitive group to act on a Steiner system S(r, k, u ) , and deduced the existence of S(5,8,24). He subsequently proved the uniqueness of this system [50]. The first part of Witt’s programme can also be used to construct MI,. However, it is not possible to deduce the existence of S(5.6,12)in quite the same way. This is because Witt identifies a block as the set of fixed points of a subgroup; in M,,, only the identity fixes a block pointwise. Nowadays, the usual procedure is to construct a combinatorial object and then deduce properties of its automorphism group. Witt’s construction has been rewritten in this spirit by Luneburg [33]. He shows that the projective plane ll of order 4 can be extended three times, in a unique manner, to give S(5,8,24).If x, y, z are the added points, then the blocks containing x, y and z must be the sets of the form {x, y, z } U L, where L is a line of Il. The blocks containing two, one or none of x, y, z meet Il in sets of 6, 7 or 8 points respectively, which are identified as geometric configurations within Il: hyperovals (complete arcs), Baer subplanes, and symmetric differences of pairs of lines respectively. (Thus, there are 168 hyperovals, falling into three classes of 56, such that the hyperovals in one class meet in 0 or 2 points, while hyperovals in different classes meet in 1 or 3 points. Each pair of the extra points x, y, z is adjoined to all the hyperovals in one class to form blocks. A similar procedure applies to Baer subplanes. Finally, each symmetric difference of two lines is a block.) Luneburg also shows that, if U is a unital in n, then U U { x , y, z} carries an
Exnemal results and configuration theorems for Steiner systems
45
S(5,6, 12). Alternatively, the latter system can be constructed by extending S(2,3,9) (the affine plane of order 3) three times in similar fashion. Given a set X of 24 points, the subsets of X form a vector space V over GF(2), where the sum of two subsets S,, S, is their symmetric difference S , A S,. Choosing the basis consisting of singletons, any subset is represented by its characteristic function, a 24-tuple of zeros and ones; the cardinality of a subset is the weight of the corresponding vector (the number of ones in it). A remarkable 12-dimensional subspace W of V was discovered by Golay [19]. It has the property that all its nonzero vectors have weight 8, 12, 16 or 24. This condition determines the numbers of vectors of each weight; for example, there are 759 = ('p)/(!) vectors of weight 8. The corresponding subsets are called octads. Now five points can lie in at most one octad; for if there were two such octads, their sum would have weight 6 or less. A counting argument now shows that any five points lie in a unique octad; that is, the octads are the blocks of S(5,8,24). There are a number of constructions for the binary Golay code W, each of which gives implicity a construction for S(5,8,24): van Lint [31] gives some of these. It is possible to find S(5,6,12) in the Golay code also. Let Y be a dodecad (a vector of weight 12 in W). For any octad B, lBAYI=8, 12 or 16, and so J Bn YI = 6, 4 or 2. Any five points of Y lie in a unique octad B, and hence in a unique set B n Y of size 6. So the intersections of size 6 of octads with Y are the blocks of S(5,6, 12). There is also a ternary Golay code, a 6-dimensional subspace of a 12dimensional vector space over GF(3), to which S(5,6,12) bears the same relation as S(5,8, 12) does to the binary Golay code. Some properties of the Steiner systems follow easily from this construction.
Proposition 2.1. (i) Let B, and B, be blocks of S(5,8,24). Then IB1n B,I = 0, 2 or 4. Moreover, i f IB,n B,I = 4, then B,A B2 is a block; and if B,nB2= 8, then the complement of B,U B2 is a block. (ii) In S(5,6,12), the complemenf of a block is a block; and if B , and B, are blocks with IB,n B,1 = 3, then B,AB, is a block. Proof. Take the sums of appropriate elements of W. Note that, strictly speaking, this proposition applies only to the systems we have constructed. Until uniqueness has been proved, we cannot assume that the result holds for all such systems; and the conclusions of Proposition 2.1 are required in the uniqueness proofs, so direct proofs by counting are still required. Some of these proofs appear in a more general context in Section 4. Another construction is related to Proposition 2.1 (ii) and the analogous fact that the complement of a dodecad is a dodecad. It produces S(5,6,12) on the union of two disjoint 6-sets, and S(5,8,24) on the union of two disjoint 12-sets. A 1-fractorizafion of the complete graph on a set A of n points is a partition of
46
P.J. Cameron
the edges (the 2-subsets of A ) into n-1 classes or 1-factors, each of which is a set of in disjoint edges (covering A ) . When n = 6, there are precisely six different 1-factorizations, any two isomorphic, and any two sharing a unique 1-factor. If X is the set of six 1-factorizations, we thus have a bijection between the set of 1-factors and the set of edges (2-subsets) of X . The correspondence is reciprocal-given an edge in A, the three 1-factors including it correspond to three edges forming a 1-factor in X.Furthermore, given a 3-subset of X , removal of the corresponding threel-factors in A leaves a disjoint union of two triangles; and reciprocally. So 3-subsets of X correspond to partitions of A into two bets, and reciprocally. Now the following sets are the blocks of S ( S , 6 , 12) on A U X : A, X ; A - { a , b } U { x ,y}, X - { x , y } U { a ,b } , where {a, b } is an cdge of the 1-factor {x. y ) (and reciprocally); A - { a , b, c}U { x , y. z } , where {ab, c } is a 3-set in the partition corresponding to (x. v, z } (and reciprocally). I n group-theoretic terms, there is an outer automorphism of the symmetric group S6. under which a transposition is mapped to a product of three transpositions, and a 3-cycle to a product of two 3-cycles. The above construction is conveniently written in terms of this automorphism. Now let A* be the set of 12 points carrying S(S,6, 12). We must construct a 12-set X" carrying a reciprocal S ( 5 , 6 , 12). By Proposition 2.1, the 132 blocks fall into 66 disjoint pairs. Let R be the set of such pairs. From a graph with vertex set 0, in which { R , R ' } and {C,C'} are joined if and only if IB n Cl = 3. (Note that two blocks meet in 0 , 2 , 3 or 4 points.) This graph is isomorphic to the graph T(12) whose vertices are the 66 pairs from a set X * of size 12, two vertices adjacent if thev contain a common element of X * . It is now possible to construct an S(S.6. 12) on X * . The corresponding group-theoretic fact is the existence of an outer automorphism of M , * . A quick construction for the reciprocal pair of Steiner systems uses a Hadamard matrix of order 12, a 12 x 12 matrix H with entries f 1 satisfying HH' = 121. (Such a matrix is unique, up to permutations and sign changes of rows and columns. L.et D be the 11 x 11 matrix with rows and columns indexed by G F ( I 1 ), and ( i , j ) entry + 1 if I - j is a nonzero square, -1 otherwise. Then H is obtained by bordering D with a row and column of + l s . Note in passing that the rows of H, taken m o d 3 , span the ternary Golay code.) Given two rows of H , there are by definition six columns where they agree and six where they disagree; call these 6-sets blocks. These 66 pairs of blocks constitute S(5,6, 12) o n the set of columns of H . Since also HTH= 121, a similar construction gives the reciprocal system on the rows of H . The bijection between the pairs of points of one system and the disjoint pairs of blocks of the other is apparent. Given four points of A * , the four blocks containing them meet pairwise in four points, so t h e corresponding four point-pairs in X* are disjoint; the complement of their union is thus a 4-subset of X * . The reciprocal construction leads back to
Extremal results and configuration theorems for Steiner systems
47
the original 4-set in A * . So there is a bijection between 4-subsets of A* and X * . Now the blocks of S(5,8,24) on A*, X* are of three types: a block of A* and the corresponding point-pair in X * ; a block of X * and the corresponding point-pair in A * ; the union of corresponding 4-subsets of A * and X * . The sequence of constructions breaks off here, since has no outer automorphism. Finally, mention should be made of Todd’s list [47] of all the blocks of S(5,8,24), Curtis’ “miracle octad generator” or MOG [ 131, and Mason’s construction using a dodecahedron [34].
3. Bounds and extremal systems Let S be a Steiner system S ( t , k, u); we almost always assume t < k < u. The diuisibility conditions, asserting that (f-:) divides (y-:) for 0s is t, are well known. An ith contraction of S is obtained by taking those blocks which contain a given set of i points, and deleting those points. It is an S ( t - i, k - i, u - i); the number of blocks in such a system is simply the quotient of the numbers occurring in the divisibility conditions. Suppose t = 2 . By hypothesis, there is a block €3 and a point x $ B . For any y E B, there is a unique block By containing x and y ; the blocks By, as y runs through B,are all distinct. Since x lies in just ( u - l ) / ( k - 1) blocks altogether (the number of blocks in the first contraction of S), we have u - 1 2 k ( k - 1). For arbitrary t 2 2, this bound can be applied to the ( t - 2)nd contraction of S, yielding
Proposition 3.1. The parameters of S ( t , k, u ) , with 2 S r < k < u, satisfy u - t + 12 ( k - t + 2)(k - t + 1). When does equality hold in Proposition 3.1? Inspection of the proof shows that, for t = 2, we have u - 1 = k(k - 1) if and only if every block containing x meets B. Since x and B were arbitrary, this is equivalent to the assertion that any two blocks meet. Such a system is called a projectiue plane. Putting k = n + 1, we have u = n 2 + n + 1; n is called the order of the plane. We note in passing that the dual of a projective plane, obtained by interchanging the labels “point” and “block”, is a projective plane of the same order.
Proposition 3.2. Suppose S(t, k, u ) attains the bound of Proposition 3.1. Then (i) i f t = 2, then S is a projective plane; (ii) if t > 2 , then ( t , k , u ) = ( 3 , 4 , 8 ) , (3,6,22), (4,7,23), (5,8,24), or (3, 12, 112). Proof. We have seen (i) already. Suppose t = 3 and the first contraction is a
48
P.J. Cameron
projective plane of order n. Then k = n + 2, u = n 2 + n + 2 . The divisibility conditions assert that (n;Z) divides (”*+;+*), that is, n + 2 divides (n’+ n +2)(n2+n + 1). By the Remainder Theorem, n + 2 divides 12, whence n = 2, 4 or 10. The divisibility conditions also show that S(4,5,9), S(6,9,25) and S(4, 13, 113) do not exist. This completes the proof. Systems with the first five parameter sets in Proposition 3.2 (ii) are unique (up to isomorphism). The existence of S(3, 12, 112) is undecided. We refer to [ l l ] for several variations on Proposition 3.2. The case t = 2 of Proposition 3.1 is a special case of a more general result. It asserts that there are at least u blocks, with equality if and only if any two blocks meet in a point. The generalization, known as Fisher’s inequality, holds in any 2-design (a collection of blocks or k-subsets of a u-set, any two points lying in just A blocks), with 2 s k s u - 1. Fisher’s inequality asserts that such a design has at least u blocks, with equality if and only if any two blocks meet in A points. (The matrix D of Section 2 is the incidence matrix of such a design, with u = 11, k = 5, A = 2.) Fisher’s inequality has been generalized by Ray-Chaudhuri and Wilson [41]. They showed that a t-design with t = 2s and t s k =su - s has at least (1) blocks. They also characterized the case of equality by intersection properties of blocks. It is worth noting that, up to complementation, the only known design meeting the bound with s 3 2 is the Steiner system S(4,7,23).However, Gross [20] has shown that the Ray-Chaudhuri and Wilson inequalities give no new nonexistence results for Steiner systems, and that no other Steiner system attains the bound. We will discuss his work further in the next section. Another inequality connecting the parameters of a Steiner system is the following.
Proposition 3.3. The parameters of S(t, k, u ) , with 1 s t < k < u , satisfy u s ( t + I)(k - t + 1). Proof. Let B be a block, x,, . . . ,x, E B, and x,, a point outside B. Then no block contains {x,. . . . , x,+,}; so the r + l blocks B,,. . . , B,,, containing t of these points are all distinct. Moreover. any two such blocks have t - 1 of the points x,, . . . , x,+, in common, and so share no further point. So B,,. . . , B,,, include ( t + l)(k - t ) points other than xI, . . . , x , + ~ giving , the desired inequality. As a simple corollary of Proposition 3.3, we note that u 3 2 k , with equality only if t = l or k = l + l . When can the bound of Proposition 3.3 be met? If u = ( t + l ) ( k - t + l), then the proof shows that any additional point lies in one of the blocks B,,. . . , that is. given a n y t + 2 points, there is a block containing at least t + 1 of them. Known systems meeting the bound with t > 1 are S(3,4,8), S ( 5 , 6 , 12) and S(5,8,24). It is conjectured that, apart from the unusual case u = 2 k (to be discussed further in the next section), the only system meeting the bound is S(5,8,24). Some information on this question follows.
,
Extremal results and configuration theorems for Steiner systems
49
Let B be a block of S ( t , k , v ) , where u = ( t + l ) ( k - t + l ) and l < t < k - 1 . Given x, y $ B and zl, . . . , z, E B, there is a unique block containing t + 1 of the points x, y , zl,. . . , 2 , ; this block must contain x and y . So the sets B-B’, where 1B nB’(= t - 1 and x, y E B’, form a Steiner system S ( k - t, k - r + 1, k ) on B, where x and y are fixed. Also, the sets B ’ - B , where B’ contains t - 1 of the points zl,. . . , z,, form a Steiner system S(2, k - t + 1, v - k ) on the complement of B. This system is resolvable: the blocks containing a given t - 1 of zl, . . . , z, form a parallel class. There exist sets X of t + 4 points, no t + 2 of them in a block. Any t + 2 of these points have the property that a unique block contains t + 1 of them. Thus the sets X-B, where I B n X l = r + l , are the blocks of an S(2,3, t+4) on X. These Steiner systems give further divisibility conditions. For example, the ( k - t - 2)nd contraction of the first system is an S(2,3, t + 2) (note k 3 f + 2 by assumption); so t = 5 (mod 6). The analysis produces, as a by-product, further insight into the rich structure of S(5,8,24). 4. Intersection triangles
Given a block B of a Steiner system S(t, k, v ) , and a set I of i points of B, the number of blocks B’ for which B nB’ = I depends only on i, t, k and v. The easiest proof of this fact includes an algorithm for computing this number. The intersection triangle of S ( t , k, u ) is a triangular array { p i j10 < j s is k } defined by the rules
and pi+l,i + p , + l , , +--lpij for O s j s i s k.
It is easily proved, by induction on i - j , that this is a good definition. Furthermore, the same proof shows the following: if B = {xl,. . . , x,,} is a block, then pijis the number of blocks B’ for which B ’ n { x l , .. . , x i } = { x I , . . . , xi}. This establishes the assertion. As an example, we give the intersection triangle for S(5,8,24).
759 506
253 176 330 210 120 56 80 40 130 78 52 28 12 46 32 20 8 30 16 16 4 4 30 0 16 0 4
77 21
5
16
1
4 4 0
1
0 0
1
0 0
0
1
50
P.J. Cameron
Our discussion of the intersection triangle is based on Gross [20]. It is clear that ps = 0 whenever t S j S i : these are the trivial zeros. In the triangle for S(5,8,24), p8,1and pLs,3 are nontrivial zeros, providing the information that no two blocks
meet in 1 or 3 points, a fact remarked on in Proposition 2.1. By Proposition 3.3, S(t, k, v ) satisfies u a 2 k , with equality only if either t = 1 or k = t + 1. The first case is trivial. For the parameters S( t, t + 1 , 2 t + 2), the intersection triangle is symmetric about its vertical axis. The reflection of p k k = 1 yields p k o = 1, or (since v = 2k) the complement of any block is a block. Moreover, the reflection of the trivial zero p k t gives a non-trivial zero P k 1 : no two blocks have just one common point. We see from this that any S ( t - 1, t, 2 t + 1) is uniquely extendable, if at all. For, in making the extension, we must add the new point to all existing blocks, and then include the complements of existing blocks as new blocks; this completes the system. Moreover, this construction always produces S(t, t + 1 , 2 t + 2 ) . So every S ( t - 1, t, 2 t + 1) is uniquely extendable. Systems with the parameters S(t, t + 1 , 2 t + 2 ) are known only for t = 3 and 5 . In any such system, t + 2 must be prime. (For, by the divisibility conditions, i ! divides ( t + 3)(t+ 4) * * * ( t + i + 1) for 2 < i < t + 1; so no prime less than t + 2 can divide t +2 . ) Mendelsohn and Hung [37] have shown that t = 9 is impossible; other values are undecided. There are some nonexistence proofs under additional assumptions, e.g. [2]. The entries in the intersection triangle are all non-negative integers. Does this place any further restrictions on the parameters t, k , v? Also, which systems have a non-trivial zero? These questions were considered by Gross [20]. First, if the divisibility conditions are satisfied, then all p,# are integers; by induction, all ptl are integers. Next, note that pk,r-= ( v - t - 1)/(k - t - 1)- 1> 0. The number p k , r - 2 is particularly interesting. It is the number of blocks disjoint from a given block in the ( t - 2)nd contraction. Now the inequality v - t + 1 3 ( k - t + 2)(k - t + 1) of Proposition 3.1 is equivalent to the non-negativity of P k . r - 2 , and Proposition 3.2 describes the systems with p k , l - 2 = 0 . Gross showed that, from the divisibility conditions and the inequality of Proposition 3.1, it is possible to deduce that every entry in the triangle is a non-negative integer; so no new inequalities are obtained. Obviously the inclusionexclusion principle provides a formula for pS as an alternating sum of the p,, with appropriate coefficients; but such a form is not well adapted for proving nonnegativity. The Heart of Gross' argument is the formula
For example, it is immediate that pij> 0 if j f t (mod 2). The formula is proved, as usual, by induction on i - j. Gross also proves the following result.
Extrernal resulfs and configuration theorerris for Steiner systems
51
Proposition 4.1. The only Steiner systems S(t, k , u ) , for 2 s t < k < u, with nontrivial zeros in their intersection triangles, are the following: (i) those of (3.2), with pk,,..*= 0 ; (ii) s(4, 7,23) and s(5,8,24), with P k , , - d = 0; (iii) S( t, t + 1, 2t + 3), with Pk.0 = 0 ; (iv) S(t, t + 1,2t + 2), with P k . 1 = 0 . In particular, the only Steiner systems in which any two blocks meet are projective planes, S(4,7,23), and S(t, t + 1,2t + 3). For related results, see Hubaut [25], Noda [39]. A slight variant of the argument which began this chapter proves a stronger result: given a block B of S ( t , k , a ) , an i-subset I of B, and a point x$ B. the number of blocks B' with x E B' and B n B'= I depends only on i, r, k , u. This number is clearly p k i ( k- i ) / ( u - k ) ; it can also be found from an intersection triangle, as we illustrate for S(5,8,24). 253 176 77 120 56 21 80 40 16 52 28 12 4 33 19 9 3 21 12 7 2 1 15 6 6 1 1 15 0 6 0 1 0
5 1
0
1
0 0
0 0
0
0 0
0
The analogous result for two or more points outside B is false in general; but it would be interesting to know in which systems it holds with i = 0. We will have a little more to say about this later. Intersection triangles can be used in other situations too. Fow example, the following triangle shows that, if X is a set of 5 points of S(3,4, 10) containing no block, then the complement of X also contains no block.
30 18 12 10 8 4 5 5 3 1 2 3 2 1 0 0 2 1 1 0 0 See also Conway [12].
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P.J. Cameron
5. Projective and a5ne spaces The subject-matter of this section and the next is “classical” geometry; we merely outline some of the theory in the context of Steiner systems. We are also concerned with embedding Steiner systems in the classical ones. Let F = GF(q) denote the finite field with q elements, and V a vector space of rank d + 1 over F ( d 2 2). Define a configuration whose points are the subspaces of rank 1 of V, and whose lines are the subspaces of rank 2, incidence being defined by set-theoretic inclusion. Clearly any two points lie on a unique line. Easy counting arguments show that we have a Steiner system S ( 2 , 4 + 1, (4‘‘ ’ I - l)/(q - 1)). We call this the d-dimensional projective geometry over F, and denote it by PG(d, 4). Note that PG(2, q ) is a projective plane of order 4. If x and y are distinct points, we denote the line joining them by x y . If W is any subspace of V, of rank e + 1, then the points and lines contained in W form PG(e, q ) (degenerate if e d 1). In particular, three non-collinear points x , , x2. x3 are contained in a projective plane 11. If L is a line meeting x , x 2 and x I x 3 but not containing x , , then L G 11, and so L meets x z x J . This property virtually characterizes projective geometries:
Theorem 5.1. Let S ( 2 , k , v ) ( 2 < h < v ) have the property that a line meeting two sides of a triangle (not at a vertex) must meet the third side also. Then either (i) k - 1 is u prime power q. and S = PG(d. q) for some d b 3 ; or (ii) u = k ’ - k + 1 and S is a projective plane of order k - 1. An alternative form of this theorem is often convenient:
Theorem 5.1’. The conclusions of Theorem 5.1 hold i f any three non-collinear points lie in u projective plane S ( 2 , k , k ’ - k + 1). For a proof, see Veblen and Young [48]. Consider a subspace PG(d - 1 , 4 ) in PG(d, 4). The rank formula for subspaces of a vector space shows that any line not contained in PG(d - 1, q ) meets it in a unique point. So these lines are the blocks of S ( 2 , 4 , q J ) on the complement of PG(d - 1.4). This system is the d-dimensional afine geometry A G ( d . q ) . Two lines of A G ( d , 4 ) are said to be parallel if they meet the “hyperplane at infinity” PG(d - 1. q ) in the same point. Parallelism is an equivalence relation satisfying Euclid’s postulate: through any point x there is a unique line parallel to a given line L. An alternative description of A G ( d , q ) is as follows: t h e points are the vectors o f a vector space of rank d over F; the lines are the cosets of subspaces of rank 1 : two lines are parallel if they are cosets of the same subspace. An ufine plane of order n is an S ( 2 , n, n’). (Thus AG(2, q ) is an affine plane of
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order q.) Call two lines of an arbitrary affine plane parallel whenever they are equal or disjoint. (This agrees with the previous definition for A G ( 2 , q).) If x # L, then x lies on n + 1 lines, n of which meet L ; so x lies on a unique line parallel to L. Thus, parallelism is an equivalence relation satisfying Euclid’s postulate. Now let S be a system S(2, k , u ) in which any three non-collinear points lie in an affine plane S(2, k, k’). We may call two lines parallel if they are contained in an affine plane and are parallel there. Then Euclid’s postulate holds; but this parallelism may not be an equivalence relation. In fact, we have the following:
Theorem 5.2. Let S(2, k, u ) (2< k < u ) have the properties (a) any three non-collinear points lie in an afine plane; (b) parallelism (defined above) is an equivalence relation. Then either (i) k is a prime power q, and S is AG(d,q ) for some d 2 3; or (ii) S is an afine plane S(2, k , k’). For proofs see Lenz [30], Hall [23]. Hall gives a class of examples to show that condition (b) is necessary. In view of this, the following result of Buekenhout [6] is remarkable.
Theorem 5.3. Condition (b) of Theorem 5.2 may be deleted if k > 3. Thus, Hall’s examples are Steiner triple systems. It follows from a result of Bruck and Slaby [4] (on commutative Moufang loops of exponent 3) that in such systems u is a power of 3. The smallest example has u = 81. Note that affine geometries, as we have defined them, are trivial when q = 2. In this case, Steiner systems can be defined by using planes (instead of lines) as blocks. We can do this most economically by letting AG,(d, 2) ( d 2 3) have as points the vectors of a vector space of rank d over GF(2), and as blocks the quadruples of vectors with sum 0. It is an S(3,4, 2d).Note that, if B, and B, are blocks of AG,(d, 2) with IB, nB,I = 2, then B,AB, is a block (cf. Proposition 2.1). This property characterizes AG,(d, 2) among S(3,4, u ) s : see Lenz [30], or Corollary 7.3. We can translate Theorem 5.1 into a different form, one which suggests many open problems. See also Hubaut [25].
Proposition 5.4. Suppose S(2, k, u ) has disjoint blocks (i.e. u > k 2 - k + 1). Then S is PG(d, q ) (d a 3 ) i f and only if, given disjoint blocks B,, B, and a point x+!BIUB,, there is at most one block containing x and meeting B, and B,. Moreover, S is PG(3, q ) if and only if the same condition holds with “at most” replaced by “exactly”.
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We mention just three open problems.
(1) Find a similar characterization of A G ( d , 4). (2) What if “at most” is replaced by “at least”? (It can be shown that v = s k ( k 2 - 2 k + 2 ) , with equality precisely for PG(3,q). Moreover, if o s 2 k 2 3 k + 2, then the condition necessarily holds.) (3) Find similar results concerning common transversals to intersecting pairs of blocks.
6. Projective, ffie and inversive planes Projective and affine planes are intimately related. Given a projective plane of order n, deleting a line (the “line at infinity”) and all of its points yields an affine plane of the same order. Conversely, given an affine plane, adjoin an ideal point to all the lines of each parallel class, and an ideal line incident with all the ideal points; the projective plane is recovered. PG(2, q ) and AG(2, q ) are the classical projective and affine planes. However, there are very many non-classical planes; the exceptions in Theorems 5.1 and 5.2 are genuine. It should be noted, though, that all known planes have prime power order. For orders n with 2 s n 8. the planes are unique (except for n = 6 , where n o plane exists): for n = 9 there are at least four nonisomorphic planes; and for t i = 10 the existence question is unsettled. The strongest nonexistence theorem is due to Bruck and Ryser [5]:
Theorem 6.1 lf a projective plane of order n = 1 or 2 (mod 4) exisrs, then 11 = a ’ + h’ for some integers a and h. The classical theorems of Desargues and Pappus are “configuration theorems” which characterize PG(2, q ) and AG(2, q ) among projective and affine planes. We state these for projective planes, together with a theorem of Gleason [18], referring to Demhowski [ 151 for variations and elaborations.
Tbeorem 6.2. A project plane of order n is PG(2, q ) (with n = q ) if and only i f ,
wheneuer a , a 2 , bib, and c,cz are concurrent lines and a , b , n a , b , = z , u Ic2f~ a,c, = y. h,c, f3 b2c, = x , the points x , y, z, are collinear.
Theorem 6.3. A projective plane of order n is PG(2, q ) (with n = q ) if and only i f . whenever a , b , c , and a,h,c, are collinear triples of points and a , b , n a , b , = z, u , c z f l a,cl = y. h , c , n b,c, = x , the points x, y, z are collinear.
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Fig. 1.
Theorem 6.4. A projective plane of order n is PG(2,2‘) (with n = 2‘) if and only i f , whenever a,, a,, a3, a4 are points with no three collinear and a,a,n a3a4= z, a,a,na,a,= y, a,a,na2a3=x, the points x, y , z are collinear. The forward implications in Theorems 6.2 and 6.3 are the theorems of Desargues and Pappus: see Arthin [ 11 for the converses. There is an elementary proof that Pappus’ theorem implies Desargues’: van Lint [32] gives a proof for the case where no degeneracies occur. The converse is also true in finite projective planes, but the problem of finding an “elementary” proof remains open. The question of extendability of projective planes is virtually settled by Proposition 3.2. The situation is different for affine planes: the divisibility conditions are always satisfied by the parameters S(3, n + 1, n 2 + 1). A system with these parameters is called an inversive plane of order n. An ovoid is a set of q 2 + 1 points in PG(3, q ) , no three collinear. Any ovoid is the point set of an inversive plane of order q, whose blocks are the non-trivial plane sections. An inversive plane constructed in this way is called egglike. The classical inversive plane I ( q ) is obtained from the elliptic quadric in PG(3, q ) , the set of points spanned by zeros of the quadratic form Q(x,,
~ 2~ , 3 =) U
XI,
+ hxox, + CX: + ~ 2 x 3 ,
X ~
where the quadratic ax2+ bx + c is irreducible over GF(q). In the spirit of Theorems 6.2-6.4, there is a configuration theorem (Miquel’s theorem) characterizing the classical inversive planes, and another (the bundle theorem) which holds in all egglike planes and is thought to characterize them: see Dembowski [15]. The most important theorem is due to Dembowski [14]:
Theorem 6.5. A n inversive plane of even order n is egglike (and so n is a power of 2). This result is very satisfactory, since there are non-classical egglike inversive planes of even order. By contrast, the only known inversive planes of odd order
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are the classical ones, but the only known restriction is the Bruck-Ryser theorem (Theorem 6.1). Extensions of inversive planes are considered in Section 7. We conclude this chapter with some remarks on embeddings. Let S = S(2, K. V). and let T be a subset of S of cardinality u with property (*)
any line of S meets T in 0, 1 or k points.
Such a set is called a {u, k}-arc in S (or a u-arc if k = 2). Obviously the nontrivial intersections of T with blocks of S give T the structure of S ( 2 , k, u ) . Considering lines through a point of T, we see that ( u - l ) / ( k - 1)<(V - l ) / ( K - 1). If equality holds, then T is called complete, and has the stronger property ( * *)
any line of S meets T in 0 or k points.
If 7 is a complete arc, and x a point outside T, then the blocks through x which meet T partition it; so k divides u. For projective planes, we obtain the following result.
Proposition 6.6. A { u , k}-arc in a projective plane of order n satisfies u S nk - n + k, with equality if and only i f it is complete. If T is a complete {u, k}-arc, then n = k k ' for some integer k', and the lines disjoint from T form a complete { v ' , k'}-arc in the dual plane (the dual arc). Denniston [16] has shown that, if q is a power of 2, then PG(2, q ) possesses complete {u, k}-arcs for every k dividing 4. N o non-trivial examples are known with k odd; and m a s [46] has shown that, if q is a proper power of 3, then PG(2, q) contains no complete { u , 3}-arc. Another interesting case is that when ( u - - l ) / ( k - 1)= ( V - l ) / ( K - 1)- l.Then T has a unique tangent at each of its points. For projective planes, the following holds.
Proposition 6.7. Let T be a {c, k}-arc in a projectiue plane of order n, with v = nk - n + 1. Then k diuides 2(n - 1 ) and k ==&+ 1. Furthermore; (i) If k divides n - 1 then any point not belonging to T lies on 0 or k tangents to T, whence the tangents to T form a {v, k}-arc in the dual plane. (ii) If k does not divide n - 1, then n and k are even, and the number of tangents through a point T is congruent to k (mod k ) . If k = 2, then there is a point (the nucleus) lying on all tangents, whose addition to T yields a complete arc. (iii) If k = &+ 1, then every line meets T in 1 or k points. In Proposition 6.7 (iii), with n = m 2 , T is an S ( 2 , m + 1, m 3 + 1 ) . Such a system is called a unital. Again, there is a classical example: the set of absolute points of a unitary polarity of PG(2, q2). O N a n [40] has shown that the classical unitals have a property contrasting with Theorem 5.1: a line meeting two sides of a
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triangle and containing no vertex is disjoint from the third side. It is not known whether this property characterizes the classical unitals. As well as intrinsic characterizations, conditions for embeddability of unitals in projective planes (or, more particularly, as the sets of absolute points of polarities) would be welcome. We refer to Taylor [43], Buekenhout [8]. AG(2,3) is a unital with m = 2 , and is embedded as such in PG(2,4) (cf. Section 2). A 10-arc consisting of a conic and its nucleus (Proposition 6.7 (ii)) in PG(2,8) has as dual (Proposition 6.6) a non-classical unital with m = 3 (the smallest so-called Ree unital). Another question of the same sort concerns subsystems of projective and affine planes which are themselves such planes. The following result holds.
Theorem 6.8. (i) If a projective plane of order n has a proper subsystem which is a projective plane of order r, then either n = r2 or n 2 r2 + r. (ii) If a projective plane of order n has a proper subsystem which is an afine plane of order r, then either n = 4, r = 3, or n 2 r2 - ‘r 2 -2Proof. We prove the second part; the first is easier. Let xi be the number of points outside the subplane and lying on i lines of the subplane. Then 1 x, = n 2 + n + 1 - r 2 , l i x i = r ( r + l ) ( n + l - r ) , x i ( i - l ) x i = r ( r + l ) ( r - l ) . From these equations, 1 ( i - 1)(i - 2)xi is computable; the non-negativity of this quantity yields the result. The first part of Theorem 6.8 is due to Bruck. A subplane whose order is t h e square root of that of the plane is called a Baer subplane. Such subplanes are characterized by the property that every point of the plane lies on a line of the subplane (and dually). Examples are common: PG(2,q“) contains a subplane PG(2, q), which is a Baer subplane if m = 2. No examples with n = r 2 + r are known at present. (Ofcourse, such planes would not have prime power order.) The second part of the theorem improves a result of Ostrom and Sherk. We have seen the embedding AG(2,3) c PG(2,4); more generally, AG(2,3) c PG(2, q) for all q = 1 (mod 3), as the Hessian configuration of points of inflection on a cubic curve. The case q = 7 attains the bound in Theorem 6.8 (ii).
7. Kantor’s embedding theorems and related results Which inversive planes are extendable? The divisibility conditions for S(4, n + 2, n 2 + 2) show that n + 2 divides 60, whence (for n > 2) there are eight possible values of n. Using Dembowski’s theorem (Theorem 6.5), the values n = 10, 18,28 and 58 are eliminated, leaving n = 3, 4, 8 or 13 for consideration. The case n = 8 is especially interesting. The divisibility conditions alone merely show that S(t, f + 6, t + 62) must satisfy t s 12. Using the inequality of Proposition 3.3 we can improve this to t s 9 ; this turns out to be far from best possible. Dembowski’s
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theorem shows that S(3,9,65) is represented by an ovoid in PG(3,8). Kantor [27] has proved that a similar statement holds for the hypothetical extension S(4, 10,66): it would be represented by a set S of 66 points in PG(4,8), no four coplanar, the blocks being the nontrivial 3-space sections. Clearly two lines meeting S in disjoint pairs of points would be skew. Thus there would be 6 6 t i . 6 6 . 6 5 . 7 = 1501.5 points lying on such lines; but there are only (8’- 1)/(81 ) = 4681 points in PCi(4,8). Extending the argument, Kantor shows:
Proposition 7.1. If an inuersiue plane of order n > 2 is extendable, then n = 3 or possibly 13. In the latter case, at least one contraction i s a non-egglike inversive plane.
(Ofcourse, the inversive plane of order 3 is twice extendable.) More generally. Kantor’s result shows that if every contraction of S(t, k, u ) (for t 2 4 ) is embeddahlc as a “sufficiently large” piece of PG(t - 1, q), then the whole system is similarly embcddable in PG(t, q). Even this does not fully describe his theorem. which applies to “geometries” (more precisely, geometric lattices) with n o restrictions on the cardinalities of subspaces. We refer to Kantor [27, 281 for precise statements. For example, S(4,5, 11) and S ( 5 , 6, 12) are embedded in PG(4,3) and PG(S.3) respectively. Note the restriction t 2 4. Thus, for example, S(3,6. 22) is not embeddable in PG(3,4), even though every contraction is PG(2,4). However. the extensions of projective spaces can be characterized as the systems S ( 3 , k . L?) satisfying the following condition: (i, Zf B,. B 2 , B, are blocks for which B , n B , and B , n B , are disjoint 2-subsets of B , , then B,nBB, is empty or a 2-set.
Thus we have the following theorem and corollary:
Proposition 7.2. An S(3, k, v ) satisfying (t) must be A G 2 ( d ,2 ) , S(3,6,22) or S ( 3 , 12, 112).
Corollary 7.3. Suppose S(3,4, t.) has the property that, whenever B, and B, ure blocks with 16,nB,I = 2 , then B,AB, is a block. Then S is A G , ( d , 2). Cameron [ 101 considered a related property. (++)
With the hypotheses of (t),I€?,
n B,I s 1.
He showed that a system S(3, k , u ) satisfying (tt) has u z 2 + i ( k - 1)’(k-2). Moreover, (W)holds with u = 2 + f ( k- 1)’(k -2) if and only if the hypotheses of the following result are fulfilled.
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Proposition 7.4. Suppose S(3, k, u ) has the property that, given any block B, the blocks meeting B in two points form an S(2, k - 2, u - k ) on the complement of B. Then S has parameters S(3,5,26), S(3,23,5084) or S(3,105,557026). The existence of these systems is undecided; none of the known S(3,5,26) satisfy the hypothesis, and systems with the other parameters are unknown. The hypothesis, reminiscent of the relation between projective and affine planes, suggests variations. For example, S(4,7,23) has the property that, for any block B, the blocks meeting B in three points form an S(3,4, 16) on the complement of B (actually AG2(4, 2)). Does such a property characterize it among S(4, k , u)s? What about other values of t‘! Compare also the remarks at the end of Section 4.
8. The symmetric difference property Motivated by Proposition 2.1 and Corollary 7.3, we say that a Steiner system S ( t , k, u ) , with k even and k < 2 t , has the symmetric difference property (SDP) if, whenever B, and B2 are blocks with IB2nB21= $ k , then B , A B , is a block. (The conditions on k and f are obviously necessary to ensure that such a pair of blocks exists, and are also sufficient, by Proposition 4.1.) It is striking that both S ( 5 , 6 , 12) and S(5,8,24) have the SDP. The only other known systems with this property are the AG,(d,2). A complete classification is not yet known, but the most dramatic result is due to Cameron [9].
Theorem 8.1. The only S ( t , k , u ) , with k A G , ( d , 2) and S(5,8,24).
= 22 - 2 > t,
having the SDP, are the
A feature of this theorem is the proof, which uses the determination of perfect binary error-correcting codes in an unexpected way: the code is a relation module for a certain group constructed from the Steiner system. At the same time, a characterization on the contractions is obtained. Though the hypotheses appear complicated, they should be compared with those of Theorem 5.1: the cases t = 2 of Theorem 8.2 and k = 3 of Theorem 5.1 are identical.
Theorem 8.2. Suppose S = S ( t , k , u ) , with k = 2t - 1, has the property that whenever p and p‘ are points and Q,, . . . , Q4 are ( t - 1)-sets for which { p } U Q, U QZ, {p’}U Q, U Q, and {p’}U QZU Q4 are blocks, then { p } U Q, U Q4 is a block. Then S = PG(d, 2) or S(4,7,23). The only further results known about the SDP are also due to Cameron [ lo], and are summarized below.
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Proposition 83. Suppose S ( t , k, u ) , with k even and k < 2t, has the SDP. Then (i) if k = 2t -4, or if k = t + 1 with t > 3, then k = 2 (mod 4); (ii) if t = 5 and k = 6 , then u = 12. Of this, only (ii) is non-trivial. The proof contains a uniqueness proof for S(5,6, 12) based on the construction via 1-factorizations of K6 (see Section 2). Since Theorem 8.1 and Proposition 7.2 have a common specialization (Corollary 7.3), it is natural to look for a common generalization. Such a result was found by Cameron [ll]. Somewhat surprisingly, the conclusions are just the disjunction of those of Theorem 8.1 and Proposition 7.2.
Theorem 8.4. Suppose S = S ( t , k, tl), with k 3 2t - 2, t 2 3, has the property that, whenever B , , B,, B, are blocks for which B1n B , and B , n B , are disjoint ( t - 1)subsets of B,,then IB, n B,I = 0 or t - 1. Then either (i) t = 3, S is AG,(d, 2), S ( 3 , 6 , 2 2 ) or S(3,12,112); or (ii) k = 2t - 2, S is AG,(d, 2) or S(5.8,24).
Proof. Note that the conditions
t 3 3 and k 3 2t - 2 avoid the possibility that the hypotheses are vacuously fulfilled. A simple counting argument shows t = 3 or k = 21 - 2; now Proposition 7.2 and Theorem 8.1 apply. Theorems 5.1 and 8.2 also have a common specialization, but no common generalization is known.
9. Further directions
In this section, some other lines of investigation will be briefly mentioned. A feature of t h e projective and affine spaces is that they are equipped with large numbers of subspaces. Teirlinck [45] and Doyen, Hubaut and Vandensaval [ 171 have studied the analogues of hyperplanes in arbitrary Steiner triple systems. The existence of hyperplanes (with properties like those in projective and affine spaces) is closely related to the rank (mod 2 or 3) of the incidence matrix, and so to the considerations of coding theory. Hall [24] and Teirlinck [44] have looked at planes, generalizing Theorems 5.1' and 5.2 by allowing different types of planes to occur. Buekenhout [7] has introduced a general theory of subspaces in "block spaces". with potential applications to designs (and Steiner systems in particular). Another generalization starting with the same observation is the theory of perfect rnatroid designs [38]. These are geometries with subspaces of various dimensions, in which an i-space and a point outside it determine a unique ( I + I)-space. and the cardinality of a subspace depends only on its dimension. (In S(I. k , v ) , the i-spaces for i < t - 1 are the ( i + 1)-sets, while the hyperplanes are the blocks.) For example, a 3-dimensional perfect matroid design consists of an S ( 2 , k , t i ) , whose blocks are the lines, with' a collection of subsystems called planes, three non-collinear points lying in a unique plane, and any plane having
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cardinality p. If k = 2, it is simply an S(3, p, u ) . For k > 2 , the cases p = kZ- k + 1 and p = k 2 are covered by Theorems 5.1’ and 5.3. Two open problems: Are there any examples with k > 3 apart from projective and affine spaces? If any two planes meet, must the geometry be a projective 3- or 4-space? Steiner systems have been coordinatized in various ways. On the one hand, the familiar introduction of coordinates in PG(2, q ) has been generalized to arbitrary projective planes; the system of coordinates is a “planar ternary ring”. (See Hall [21], Dembowski [15].) On the other, there is a familiar connection between Steiner triple systems and certain varieties of quasigroups (or loops). (The exact method of coordinatization depends on whether we want PG(d, 2) or AG(d, 3) to be coordinatized by the additive group of the underlying vector space.) This too has been extended to other systems, leading to a close relation between combinatorics and universal algebra: see Ganter and Werner’s and Quackenbush’s article in this volume. In all cases, structural properties of a Steiner system are reflected by algebraic properties of the coordinatizing structure. If the blocks of S(2, k, u ) are regarded as the vertices of a graph, two vertices adjacent if the corresponding blocks intersect, then the resulting graph is “strongly regular” in the sense of Bose [3]. That author has also given sufficient conditions (involving only the parameters) for a strongly regular graph to be obtained from a Steiner system (and, more generally, from a “partial geometry”). A good example of the connection between structural properties of graph and Steiner system occurs in the work of Sims [42] on graphs satisfying the 4-vertex condition. Cameron [lo] investigated similar connections between Steiner systems with t = 3 and “association schemes” (generalizations of strongly regular graphs). For r>3, little is known except an isolated result of Ito and Patton [26] characterizing S(4,5, 11). An important invariant of a Steiner system is its automorphism group. Mendelsohn [36] has shown that any group is the automorphism group of a Steiner triple or quadruple system. Much work has been done on finding, for example, Steiner triple systems with prescribed automorphisms. From the point of view of this article, the most interesting systems are those with large automorphism groups and, in particular, those S(r, k, u ) which have f-fold transitive groups. The systems S(5,8,24) and S(5,6,12) and their contractions, together with the projective and affine spaces, classical inversive planes, (and higher-dimensional “circle geometries”), and classical and Ree unitals, all fall into this category, and it is believed that there are no others. However, the limitations of our knowledge are shown clearly by the fact that not even the doubly transitive Steiner triple systems have been determined! Hall [22] has shown that a Steiner triple system whose automorphism group is transitive on non-collinear triples of points must be PG(d, 2) or AG(d, 3). (This is shown by identifying the planes as projective or affine.) Other results in this vein are surveyed by Kantor [29]. A large automorphism group often forces a system to have structural properties of the kind discussed in this paper.
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References [ l ] E. Artin, Geometric Algebra (Interscience, New York, 1957). [ 2 ] E.F. Assums Jr. and M.T. Hermoso, Non-existence of Steiner systems of type S(d - 1, d, Zd), Math. Z. 138 (1974) 171-172. [3] R.C. Bose, Strongly regular graphs, partial geometries, and partially balanced designs, Pacific J. Math. 13 (1963) 389-419. [4] R.H. Bruck, A Survey of Binary Systems (Springer-Verlag, Berlin, 1958). [5] R.H. Bruck and H.J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. I (1949) 88-93. [6] F. Buekenhout, Une caracterisation des espaces affines basee sur la notion d e droite, Math. Z, 111 (1969) 367-371. [7] F. Buekenhout, What is a subspace? in: P.J. Cameron, ed.. Combinatorial Surveys (Academic Press, London, 1977) 1-22. 181 F. Buekenhout, Existence of unitals in finite translation planes of order q2 with a kernel of order q. to appear. [Y] P.J. Cameron, Characterisations of some Steiner systems, parallelisms and biplanes, Math. Z. 136 (1974) 31-39. [ 101 P.J. Cameron. Two remarks on Steiner systems, Geometriae Dedicata 4 (1975) 403-418. 11 11 P.J. Cameron. Extensions of designs: variations on a theme, in: P.J. Cameron, ed., Combinatorial Surveys (Academic Press, London, 1977) 23-43. Conway. A group of order 8.315,553.613.086,720.~00, Bull. London Math. Soc. 1 (1969) [12] J.H. 79-88. [13] R.T. Curtis. A ncw comhinatorial approach to M24. Math. Proc. Camb. Phil. SOC. 79 (1976) 25-42. [ 141 P. Dembowski, Mobiusehenen gerader Ordnung. Math. Ann. 157 (1964) 179-205. I151 P. Dembowski. Finite Geometries (Springer-Verlag. Berlin, 1968). [ 161 R.H.F. Denniston. Some maximal arcs in finite projective planes, J. Combinatorial Theory 6 (1969) 317-319. 1171 J. Doyen, X. Hubaut and M. Vandensaval. Ranks of incidence matrices of Steiner triple systems, Math. Z. 163 (1978) 251-259. [lX] A.M. Gleason. Finite Fano planes, Amer. J. Math. 78 (1956) 797-807. [I91 1M.J.E. Golay. Notes on digital coding, Proc. I R E 37 (1949) 657. 1201 B.H.Gross, Intersection triangles and block Intersection numbers for Steiner systems, Math. Z . 139 (1974t 87-104. 1211 M. Hall Jr.. Projective planes, Trans. Amer. Math. SOC.5 4 (1943) 229-277. 1221 M. Hall Jr.. Group thcory and block designs. in: L.G. Kovacs and B.H. Neumann. cds.. Proc. Intern. Conf. Theory of Groups (Gordon & Breach, New York, 1967) 11.3-134. 1231 M. Hall Jr.. Incidence axioms for affine geometry, J. Algebra 21 (1972) 535-547. [24] J.1. Hail. Steiner systems with geometric minimally generated subsystems, Quart. J. Math. Oxford ( 2 ) 25 (1974) 41-SO. [25] X. Huhaut. Systemes de Steiner minimaux, Bull. SOC.Math. Belgique 2 3 (1971) 411-415. [26] N. Ito and W.H. Patton. On a class of 4-(c,5, 1) designs, to appear. [27] W.M. Kantor. Dimension and embedding theorems for geometric lattices, J . Combinatorial Theory ( A ) 17 (1974) 173-195. 1781 W.M. Kantor. Envelopes for geometric lattices. J. Combinatorial Theory ( A ) 18 (1975) 12-26, [29] W.M. Kanror. ?-transitive designs, in: M. Hall Jr. and J.H. van Lint, eds.. C'omhinatorics ID. Keidcl. Dordrecht, 1975) 365-418. 1301 H . Lcnz. Z u r Bcgrundung der analytischen Geometrie. Sitz.-Ber. Bayer. Hayer Akad. Wiss. I 1054) 17-71. [31] J . H . van l i n t . Coding Theory, Lccturc Notes in Math. 201. (Springer-Verlag, Berlin, 1971). [ 3 2 ] J.H. van Lint, Combinatorial Theory Seminar Eindhovcn, 1-ccturc Notes in Math. 382 (SpringerVcrlag, Bcrlin. 1974). [ 3 3 ] H . Lunehurg. Transitive Erweiterungen endlicher Perrnutationsgruppen, Lccturc Notes in Math. S4 (Springer-Verlag, Berlin, 1969).
Extremal results and configuration theorems for Steiner systems
63
[34] D.R. Mason, On the construction of the Steiner system S(5,8,24), J. Algebra 47 (1977) 77-79. [35] E. Mathieu, Sur la fonction cinq fois transitive de 24 quantitts, J. Math. Pures Appl. (Liouville) (2) 18 (1873) 24-46. [36] E. Mendelsohn, On the groups of automorphisms of Steiner triple and quadruple systems, In: E. Mendelsohn, ed., Proc. Conf. Algebraic Aspects of Combinatorics, Utilitas Math., Winnipeg, (1975) 255-264. [37] N.S. Mendelsohn and S.H.Y. Hung, On the Steiner systems S(3,4, 14) and S(4,5, IS), Utilitas Math. 1 (1972) 5-95. [38] U.S.R. Murty, J. Edmonds and H.P. Young, Equicardinal matroids and matroid designs, Proc. 2nd Chapel Hill Conf. Combinatorial Mathematics, Chapel Hill, NC (1970) 498-541. [39] R. Noda, Steiner systems which admit block transitive automorphism groups of small rank, Math. Z. 125 (1972) 113-121. [40] M.E. O’Nan, Automorphisms of unitary block designs, J. Algebra 20 (1972) 495-511. [41] D.K. Ray-Chaudhuri and R.M. Wilson, On [-designs, Osaka J. Math. 12 (1975) 737-744. [42] C.C. Sims, On graphs with rank 3 automorphism groups, Unpublished. [43] D.E. Taylor, Unitary block designs, J. Combinatorial Theory (A) 16 (1974) 51-56. [44] L. Teirlinck, Planes and hyperplanes of 2-coverings, Bull. SOC.Math. Belgique 29 (1977) 73-81. [45] L. Teirlinck, On projective and affine hyperplanes, to appear. [46] J.A. Thas, Some results concerning {(q + l)(n - 1); n}-arcs and {(q + l)(n - 1)+ 1; n}-arcs in finite projective planes of order q, J. Combinatorial Theory (A) 19 (1975) 228-232. [47] J.A. Todd, A representation of the Mathieu group MZ4as a collineation group, Ann. di Mat. Pura ed Appl. (4) 71 (1966) $?9-238. [48] 0. Veblen and J.W. Young, Projective Geometry (Ginn, Boston, 1916). [49] E. Witt, Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Hamburg 12 (1938) 256-264. [50] E. Witt, Uber Steinersche Systeme, Abh. Math. Sem. Hamburg 12 (1938) 265-275.
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Annals of Discrete Mathematics 7 (1980) 65-70 @ North-Holland Publishing Company
THE PROBLEM OF THE HIGHER VALUES OF
t
R.H.F. DENNISTON Department of Mathematics, University of Leicester, Leicester LEI 7RH, England
About systems with parameters ( t , k, u ) , where t is 2 or 3, a vast body of knowledge has grown up, and is reviewed elsewhere in this volume. There is also a review by Cameron of the classical systems (5,6,12) and (5,8,24), with their contractions - another subject about which a formidable amount is known. The contrast is remarkable when we consider our ignorance about even the existence of other systems for which r is greater than 3. It is curious that Witt [ll] should have chosen to extend the name “Steiner systems” to these finite geometries. In fact, Steiner [lo] proposed a sequence of problems, the first two of which amount to the construction of a system (2,3, N) and of its extension (3,4, N + 1).However, his third problem is not, as might have been supposed, equivalent to the construction of a “Steiner system” (4,5, N + 2). And the strange thing is that, whereas Steiner had in one case solved his own fifth problem, nobody has yet constructed a ‘‘Steiner system” for which r = 6. (I am, of course, disregarding the trivial cases t = k and k = u, as well as the designs and tactical configurations for which h > 1.) We have two significant necessary conditions for existence, one being the familiar divisibility condition:
The other comes from Fisher’s theorem that a design, with t equal to 2, must have at least as many blocks as points; when we consider an extension to a higher value of r, we have the condition that u - t + l >(k-t+2)(k-
t+l).
Kramer and Mesner [6] review some attempts that have been made to strengthen these necessary conditions -but, apart from some extensions of the non-existent system (4,10,66), they find no “admissible” set of parameters that can be excluded in such a way. Positive results are the discoveries of systems (5,6,12) and (5,8,24), which (as already mentioned) can be regarded as classical; of systems (5,6,24), (5,7,28), (5,6,48) and (5,6,84) 131; and of a system (5,6,72) [S]. These all have in 65
66
R.H.F. Denniston
common the property of being invariant under a group PSL (2, u - 1). Each has, of course, a contraction for which t = 4 : but there is no case in which a system with t equal to 4 has been discovered, and its extension with t equal to 5 has not. Negative results begin with the non-existence [ l l ] of a system (4,6, 18). A computer search [8] for a system (4,5,15) was exhaustive and unsuccessful. And the non-existence of a system (4,10,66) is shown by Cameron [l, p. 351 to be implicit in work by Kantor [5]. The extensions of these designs, which have admissible parameters up to (9, 10,20) and (12, 18,74), are of course also non-existent. Another result in [ 5 ] is that, if a system (4,15,171) exists, at least one of its contractions is a non-Miquelian inversive plane of order 13: it would be surprising if this latter geometry were found to exist. General theorems about lower values of t are not usually helpful for our purpose; one exception is Dembowski’s result [2,6.2.14] that an even number, not a power of 2, cannot be the order of an inversive plane. Admissible sets of parameters, ruled out by this theorem, begin with (4, 12,102) and (4,20,326). This leaves, as we can see from the table in [6], plenty of admissible sets of parameters for which the question of existence is undecided. It may be of interest, or may at least save people from wasting effort on hopeless searches, if we consider the feasibility of methods with which this problem can be attacked. We might have hoped that exhaustive searches, in the style of [8], could be made in other cases. However, it was possible for Mendelsohn and Hung to succeed because, up to isomorphism, there are only (as was known) two systems (2,3, 13), and (as they themselves found) four systems (3,4,14). The situation in the very next case is entirely different [7]; and this seems to close the prospect of a similar method ever being used again for our problem. There remains the possibility of requiring the unknown system to be invariant under some group. It might be objected that the classical systems can be described i n various ways without their groups being explicitly mentioned, and that this raises the hope of constructing a new system by some ingenious trick. But in fact any short description of one of these big systems is sure to have, implicit in it, the existence of a good-sized group that leaves the system invariant. And it will be realistic for us to d o our research, not by hoping for a trick, but by thinking about some group from the beginning. The smallest undecided set of parameters is (4,5, 17). Here, I have considered all the primes which might possibly have served as periods for automorphisms, 2nd disposed of them by machine searches. So I can assert that, if a system (4,5, 17) exists, its automorphism group must be trivial. But, once more, there is hardly any larger case in which a very small group could be exhaustively considered; we soon find that, say, a doubly transitive group, or something even larger, will be needed if the search is to be feasible. On the other hand, if the group is too large, the hypothesis that it leaves the supposed system invariant may be easy to disprove. Let us look at some simple propositions which are often useful in this way.
The problem of the higher values of t
61
Theorem 1. Let G be a group of permutations of a u-set, and let a t-subset T be fixed by a subgroup H. Let { U } be the collection of all (k - t)-subsets disjoint from T. Then, in any system (t, k, u ) inuariant under G, there is a block T U U, where U E{U}, and U is fixed by H.
Proof. The definition of a “Steiner system” tells us that U exists. Suppose if possible that U is not fixed by H; then some permutation in H fixes T, but sends U to a different subset U*. By the hypothesis that G fixes the system, not only T U U but also T U U* is a block containing the t-subset T, and the definition is contradicted. Corollary 1. If H fixes no element of { U } , there is no system (t, k, v ) inuariant under G. Corollary 2. If H fixes just one element U , of { V } ,any system (t, k, v), inuariant under G, must include T U U1 as a block. Although I have not tried many cases, my impression is that these Corollaries will rapidly dispose of most hypotheses that large permutation groups could be applied to our problem. The best hope seems to be offered by the groups of the projective line, and particularly by the group PSL(2,v-l), which (as we have seen) has already had some successes. Even this group does not seem to work when t = 4 (that is, for the small proportion of admissible parameter-sets (4, k, u ) that have a prime power plus one as u). In fact, any 4-subset of a projective line is fixed by three involutions in the PGL group; and I find in practice (though I have not tried to prove) that the PSL group always includes at least one of these three. This means that, in the supposed system (4,k, u), every block would (by Theorem 1) be invariant under enough involutions to fix each of its 4-subsets at least once-which seems hopeless. If we move on, accordingly, to the cases where t = 5 , we come across some more propositions.
Theorem 2. No system (4,5,p ) , on the residues modulo the prime p as points, can be fixed by the dihedral group (x --j x + 1, x ---* -x>.
Proof. In Corollary 2, put T={-2, -1,1,2} and H =(x 4-x); since 0 is the only fixed point, we must have a block {-2,-1,0,1,2}. But then, since the supposed system is fixed by x + x + 1, we have another block {-1, 0, 1,2,3}. This is a contradiction, since one t-subset ( t = 4) is in two blocks. Corollary 3. If p = l m o d 4 (so that -1 is a quadratic residue), no system + 1) can be inuariant under PSL (2, p ) .
(5,6,p
R.H.F. Denniston
68
Corollary 4. In the contrary case (p =--1 mod 4), Q system (5,6, p + 1) cannot be fixed by PGL (2, p), though the possibility of its being fixed by PSL (2, p) remains open. These propositions, though I have stated them as above to fix the ideas, can be immediately extended to systems (2j- 1,2j, p” + l), where the prime p is greater than 2j. So there are various admissible sets of parameters: (5,6, 18), (7,8,30), ( 5 , 6 , 3 0 ) ,(7.8,38), . . . , for which it can be said that not even a PSL group fixes a system. Now suppose, if possible, that a system (6,7,65) is invariant under PSL (2, 26). Let i be a primitive mark of GF (2“), so that i6’ = 1. Take T in Theorem 1 to be the 6-subset (‘9
.12, ; X I
1 .1
.33
7
I-
.51
,1
.T4
, I‘
1,
and H to be ( x -+i 2 ’ x , x -+ l / x ) . Then Corollary 1 applies, since H leaves T invariant but has no single fixed point. We can use the same T and H to prove that no system (6,9,65) is invariant under the same group. In fact, the only 3-subset fixed by H is {io, i 2 ’ , i42}; so, by Corollary 2, the supposed system has a block {i” .9 . I 2 . 2 1 i”’ i 3 3 , i 4 2 , i ” , p 4 } , .I , I , , 3
1
Multiplying by i ” , we get another block i”, i 3 } .
Once more, we now have the impossible situation of two blocks with t points in common (though t is now 6). Likewise, PSL (2, 5 2 ) , which has a 6-subgroup with orbits of sizes 6, 3, and 2 (but not 1). cannot leave a system (9, 10,26) invariant. There is another line of argument by which we can rule out various possibilities of systems being fixed by groups. We know how to calculate the number, b say, of blocks in any system such as we are considering. And we know that, if it is to be invariant under a group G, the system will have to be the disjoint union of a suitable set of orbits, chosen out of the whole collection of orbits of G on k-subsets of our u-set. Now a prime p, if it divides the order of G, will also divide the cardinalities (or shall we say “sizes”) of most orbits in the collection; the only exceptions are orbits made up of k-sets, each of which is fixed by some permutation of period p in G. This may enable us to prove, by reducing modulo p, the impossibility of the assumption that b is the sum of the sizes of an appropriate set of orbits. The hypothesis of a system (4,5, 17) fixed by PSL (2, 24), though we have already seen arguments against it, may serve as an easy example to illustrate this method. One 5-subset of PG (1, 24), consisting of the point at infinity together with the subfield GF(2*) of GF(z4), has in PSL(2,z4) a stabiliser which is
The problem of the higher oalues of r
69
precisely PSL (2, 22). So we can find the size of the orbit to which this 5-subset belongs; namely,
-
17 16 15 = 68 = 3 (mod 5 ) . 5.4.3 We find that any 5-subset, if it is fixed by a permutation of period 5 in the group, must be one of these 68 (it suffices, by the Sylow theorems, to look at the three 5-orbits of a single such operation). Therefore, of the sizes of orbits on 5-subsets, all are congruent modulo 5 to 0 except the one congruent to 3. And n o combination of these sizes will add up to b, since b=(147)/(i)=22.7.17-1
(mod5).
Likewise, of the sizes of orbits of 8-subsets under PSL (2, 26), all are congruent modulo 7 to 0 except one congruent to 4. So this group does not fix a system (6,8,65), for which b = 6. For 8-subsets under PSL (2,43), two orbits have a size congruent modulo 7 to 6. It follows that this group fixes neither (5,8,44), for which b =3, nor (6,8,44), for which b = 4. And likewise, PSL (2,127), under which two orbits of 8-subsets have a size that =4 mod 7, fixes neither (4,8, 128) (b =3), (5,8,128) ( b = 2), nor (6,8, 128) (b = 5 ) . There are other cases in which the impossibility of a system, fixed by a PSL group, can be established, even though neither of the simple arguments given above will apply. One usually begins by seeing what k-subsets are forced into the supposed system by Corollary 2, and then reaches some sort of impasse after going a little further, without an exhaustive search being necessary. The fact that (7,8,20) is one such case has been established, not only by me, but independently by researchers at Preston Polytechnic [4]. Other cases are (11,12,24), (7,8,24), (7,8,26), (5,6,28), (7,9,30) and (7,8,44). I should like to conclude by answering a question that was asked after one talk I gave on the problem. Is it feasible to add one more point A to PG (1, q), and find a system (having q + 2 as its u ) which is invariant under PSL (2, q ) , the point A being fixed under all permutations? This method may have worked for other problems, but it seems hopeless for the present one. Suppose, for instance, that a system (6,7,25) could be constructed in this way: we note that PSL(2,23) includes many permutations of period 2 or 3, but that no point of PG (1,23) is fixed by any of these. So a 6-subset fixed by such a permutation must, according to Corollary 2, go into a block with the point A. But this means that, relative to A, we have a system (5,6,24), whose blocks include all the 6-subsets fixed by permutations of period 2 or 3 - and there are far too many of them. Likewise €or the possibility of extending the other systems (5,6,12u) that have been found. Finally, I have tried to extend the (unique) system (5,7,28) that is invariant under
70
R.H.F. Dennisfon
PSL ( 2 , Z3); and, even if no assumption is made that a group fixes the extension, I find that it does not exist.
References [ 11 P.J. Cameron, Extensions of designs: variations on a theme, in: Combinatonal Surveys, Proceedings of the Sixth British Combinatorial Conference (Academic Press, London, 1977) 23-43. [2] P. Dembowski, Finite Geometries (Springer, Berlin, 1968). [3] R.H.F. Denniston. Some new 5-designs. Bull. London Math. SOC. 8 (1976) 263-267. [4] T.S. Griggs, M.J. Grannell, and D.A. Parker, Personal communication. [ S ] W .M. Kantor, Dimension and embedding theorems for geometric lattices, .I.Combinatorial Theory ( A ) 17 (1974) 1 7 S 1 9 5 . [6] E.S. Kramer and D.M. Mesner, Admissible parameters for Steiner systems S(r. k . u) with a table for all (u-f)<498, Utilitas Math. 7 (1975) 211-222. [7] C.C. Lindner and A. Rma, There are at least 31,021 non-isomorphic Steiner quadruple systems of order 16. Utilitas Math. 10 (1976) 61-64, [8] N.S. Mendelsohn and S.H.Y. Hung, On the Steiner systems S(3.4, 14) and S(4.5, 15). Utilitas Math. 1 (1972) 5-95. [9] W.H. Mills, A new 5-design. Ars Combinatoria 6 (1978) 1 9 3 1 9 5 . [lo] J. Steiner, Combinatorische Aufgabe. J. reine angew. Math. 45 (1853) 181-182. [ l l ] E. Witt, Uber Steinersche Systeme, Abh. math. Sem. Univ. Hamburg 12 (1938) 265-275.
PART I11
STEINER SYSTEMS WITH GIVEN PROPERTIES, COLLECTIONS OF STEINER SYSTEMS, AND THEIR RELATIONSHIP TO OTHER COMBINATORIAL CONFIGURATIONS
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Annals of Discrete Mathematics 7 (1980) 73-87 @ North-Holland Publishing Company
DIMENSION IN STEINER TRIPLE SYSTEMS A.J.W. HILTON Depanmenr of Mathematics, The University of Reading, Whiteknighrs, Reading, England
and
L. TEIRLINCK Depomnenre Wiskunde, Vrije Uniuersiteit Brussel, Pleinlaan 2, 1050 Brusrel, Belgium
In this paper we discuss the notion of dimension in linear spaces, of which Steiner triple systems (STS)are a special case. We give in detail proofs of the existence of STS’s of various dimensions. We conclude with a brief discussion of some allied notions.
1. Dimension in linear spaces Let X be a finite set (whose elements will be called points) and B a collection of subsets of X (called lines) such that: (i) any two distinct points of X are contained in exactly one line, (ii) every line contains at least two points. Then we say that the pair (X, B) is a linear space. The cardinality of X is the order of (X, B ) . Trivial linear spaces occur when B = fl or when X is the vertex set and B the edge set of a complete graph. A Steiner triple system (STS) is a linear space whose lines each consist of three elements. More generally, any balanced incomplete block design in which any two distinct points are in exactly one block (so that A = 1) is a linear space. A subspace of a linear space (X, B) is a linear space ( Y,%) such that Y c X and V c B. The following lemma comes immediately from the definitions.
Lemma 1. If (XI,B1),. . . ,(X,, B,)are subspaces of a linear space ( X , B),then
(x,n . nx,, B,n . .n B,) is also a subspace. As a consequence of this lemma we define the intersection ( X , , B,) n * * . f l (Xr, B,) of the subspaces (XI,B,),. . . , (X,, B,)of the linear space (X, B) to the subspace ( X In * * nX,, B , n r l Br). Let Z c_ X. The intersection (Y, %) of all the subspaces which contain Z is the subspace generated by Z; it is the “smallest” subspace which contains 2. If ( Y , % )is the subspace generated by Z, then we write Y = [ Z ] and call Y the I-closure of Z; if we wish to emphasize that the
-
- -
73
A.J.W. Hifion. L. Teirfinck
74
space generated by Z is in ( X , B) we may write [Z],. Notice that in general [ Z ] + Z U { z : z belongs to a line containing two members of Z}. The operation 2 ---* [Z] is a closure operation: Z ‘ c 2 j [Z’I c [ZI,
2 c [ZI,
[ZI = “Zll.
Furthermore
The set of all subspaces of (X, B), ordered by inclusion, is a lattice. If a set { x l , . . . , x,}c X generates a subspace (Y,V), but n o subset of Y of cardinality less than e generates (Y,W) then the set {x l, . . . , x,} is I-independent (the use of the prefix 1- is to make a distinction from similar terminology used later in connection with matroids; the letter I stands for linear space). Every subset of X of cardinality 0, 1, or 2 is I-independent. The collection of all I-independent sets is denoted by 9.If a set of points is not 1-independent it is I-dependent. An I-independent set which generates (X, B) is called an 1-base for ( X , B ).
Theorem 1. Any subset of an 1-independent set is I-independent.
Proof. Let Z be a non-empty 1-independent set and let Z ’ s Z . If Z ’ is not I-independent there is a set Y such that IYJ<)Z’I and [Y]=[Z’]. Then [ Y u (Z\ 271= “Yl u [Z \ Z’ll = [[Z’IU [Z\Z’]]
= [Z].
But 1 Y U ( Z \ Z ’ ) J< J Z J ,which implies that Z is not I-independent, a contradiction. This proves the theorem. Every subspace ( Y,%) is generated by some I-independent set of points. If an I-independent set { x l , . . . ,x,} generates a subspace (Y, W)then (Y, W)has dimen%) is denoted by d ( Y, %). If x EX, sion e - 1. The dimension of a subspace (Y, then d({x}, 8)= 0; d(g, (3) = -1. If (X, B) is a linear space then an I-hyperplane of ( X , B ) is a subspace (Y, %) of (X, B) such that d ( Y, 5%) = d (X, €3) - 1.
If ( Y,W) is a proper subspace of (X, B), but there is n o proper subspace (Z, D ) of ( X , B) of which (Y, W)is itself a proper subspace, then (Y, W) is an 1-hyperspace of ( X , B). As we shall see later on, it is possible for the dimension of an I-hyperspace of ( X , B ) to be less than, equal to or greater than the dimension of ( X , B )itself. The only restriction satisfied in general by the dimension of an I-hyperspace is given in the following lemma.
Dimension in Steiner triple systems
75
Lemma 2. Let (Y,’&)be an 1-hyperspace of ( X , B). Then d ( Y,V) 3 d ( X , B ) - 1.
Proof. Let d ( Y , V )= y, suppose {al,. . . , ( X \ Y ) .Then ia1,. .
. ?
%+I,
generates ( Y , V ) and let b E
b)
generates a subspace (Z, D ) which contains ( Y ,V) as a proper subspace. Since (Y, 5%’) is an I-hyperspace, (Z, D) = ( X , B),and so d ( X , B)=sy + 1, which proves the lemma. If each I-hyperspace of each subspace of ( X , B) has dimension one less than the dimension of the subspace itself, and so is also an I-hyperplane of that subspace, then the space (X, B) is non-degenerate. Otherwise ( X ,B ) is degenerate. If ( X ,B ) is degenerate, but each hyperspace of each subspace of ( X , B) has dimension not greater than the dimension of the subspace itself, then the space (X, B) is slightly degenerate. If ( X , B ) is degenerate and not slightly degenerate it is uery degenerate. The dimension drop of ( X , B ) is max(d(Z D )- d ( Y ,V)), where (Y, V) is a subspace of ( X , B) and ( Z , D) is a proper subspace of (Y, V). Thus the dimension drop of a non-degenerate space is -1, of a slightlydegenerate space is 0, and of a very degenerate space is al. We remark that if (Y, V) is a subspace of ( X ,B)then it is possible to replace the set of lines V by an alternative set of lines V’ so that ( Y , V ’ )is a subspace of ( X , (B\V) UV’). If (Y, V) and ( Z ,D ) are subspaces of ( X , B) which intersect properly (in a further subspace, by Theorem I), and then V is replaced by V’then, quite clearly, it is possible that Z n Y may no longer be the ground-set of a subspace . We also point out that it is quite possible for there to be Y c X , EX, b $ [ Y U {a}] and a E [Y U{ b}]. Teirlinck [14,151 has found examples to show that if A is an I-base for a linear space and A = A l UA, where A, nA, = 8, then it can happen that [A,] n[AP]# 9. Also if [A] = X there is not necessarily a subset A ’ c A such that IA’(= d ( X ) + 1 and [A’]= X .
2. Tbe existence problem for Steiner triple systems of given dimension As remarked earlier, a Steiner triple system (STS) is a linear space whose lines each consist of three elements. Henceforth in this paper, except where it is specifically indicated to the contrary, all linear spaces will be STS’s. Trivial STS’s (S,d)occurwithS=9P=@,orwith1S1=1 a n d d = p ) , o r w i t h ISI=3and158\=1.
76
A.J.W. Hilton, L. Teirlinck
It is well-known [8] that a necessary and sufficient condition for the existence of an STS of non-zero order is IS(= 1 or 3 (mod 6). We shall call positive integers =1 or 3 (mod 6) admissible. A problem raised initially by Szamkolowicz [ 10- 131, and studied and extended by Doyen [2, 31. Hilton [5] and Teirlinck [14, 151 is to determine for which n and d there exists an STS (S, d)of order n and dimension d. A further requirement is that (S, d)should be non-degenerate, slightly degenerate, have dimension drop 1, etc. The only cases which are fully determined occur when d = 2, i.e. (S, Se) is a plane. In that case Doyen ([2], [3]) proved that there is a non-degenerate plane for all admissible orders 3 7 , and that there is a degenerate plane for all admissible orders 2 1 s . If d 3 3, so that ( S , d)is a space, then Hilton [ 5 ]proved that there is a space of dimension d for all sufficiently large admissible n. The particular case d = 3 has received much attention. It was proved by Doyen [3] and Teirlinck [15] that there are no spaces of orders n < 15, n = 19, 21, 25, 33, 37. Teirlinck [15] has also shown that the only spaces of order <31 are P ( 3 , 2 ) and A(3,3), the projective space of dimension 3 over GF(2) and the affine space of dimension 3 over GF (3) respectively, and he has also classified the spaces of dimension 3 and order 31. The construction techniques of Hilton [ 5 ] and Teirlinck 114, 1 S ] show that there are spaces of all other orders, except possibly 43, 51, 67, 69, 139. 141 and 145 which are, at the moment, undecided (this list of figures may rapidly become out of date). We shall describe in the next few sections some of the more important methods used in tackling this existence problem as well as some of the more important results.
3. Non-degenerate planes of order 6 k + 3 In this section we give Doyen's construction [2] for non-degenerate planes of order 6 k + 3 ( k z 1). Let G,,, G I , G , be three pairwise disjoint copies of a multiplicative abelian group of order 2 k + 1. For x E G let xi ( i = 0 , 1 , 2 ) denote the corresponding element of G,. Let &, bl,6,be any three permutations of the elements of G. W e shall construct an STS ( S * , d*). where S* = G,,U G , U G,. The set .PI* consists of (i) all triples {x,, xl, xz}, where x E G (these are called the uerficals of the system). (ii) all triples { x , , y , , ~ , + ~where }, i=O, 1,2; x , y , z ~ G x; # y ; and 4 i ( x ) q 5 , ( y ) = [d4 ( Z I T . It is easy to see that ( S * , d*)is an STS if we bear in mind that each element of a multiplicative abelian group of odd order has exactly one square root; ( S * , d*) is said to be deriued from G modulo &, c#J,,&.
Dimension in Steiner triple systems
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Lemma 3. If (T,B) is a proper subsystem of (S*, d*)of order 3 7 , then the vertical through any point of T is contained by T. Proof. Since (TI37 it is easy to see that TnGi#9(i=0,1,2). Put ITnGi(= ti (i = 0, 1,2). We may suppose that t23max (to, tt). Suppose there is a point x1 E T n G, which is not on any vertical through any of the points of T f lG2.In that case, each of the triples containing x1 and a point of T n G2 contains also a further point of T r l G,. It follows that t , - 13tz. This is a contradiction. It follows therefore that each point of T n G1 is on a vertical which is also on a point of T n G2. Since the vertical through each point of T n GI contains a point of T n G2 it also contains a point of T n Go. With the indices being taken modulo 3, if is a point of T f lGi-, which lies of T n Gi then, by the construction, the triples on a vertical through a point {+,, yi-l, zi}, with yi-t E T n Gi-, and zi E T n Gi, provide an injection from (Tn Gi-l)\{q-l} into ( T n Gi)\{&}. Thus 4 2 ti-1 ( i = 0, 1,2), and so to= rl = t2. It now follows that the verticals through the points of T n G1 include all the points of T. This proves the lemma. Lemma 4. Let A be a subset of a finite abelian group G and let 4 be a permutation of the elements of G. Suppose that, for each x, y E A, x # y, there exists a z E A such that 4(~)4(y)={+(z)}'. Then &(A) is a coset of some subgroup of G. Proof. Let yo be a given element of A. Then, since G is a group, I{+(x)&(y,): x E A)\=IAl. Also I { [ ~ ( Z ) ~ : Z E A } I ~ IBut A I .since { 4 ( ~ ) 4 ( y ox) E A ) c { 4 ( ~ ) 4 ( X, ~ )Y E A ) C {[4(z)P: z E A ) it follows that
I{[@(z)p: z E A)] = !A\. Therefore
{4(x)4(yo):x E A) = {[4P :z E A ) = {4(x)4(y): x, Y E A). Now let A * ={4(x)[4(yO)]-' : x E A). Then the unit element of G is in A*. Also, if x, w E A, then 4(X)[4(Y 0)l- 4(w)[ 4(Yo)l= +(x)4(w)[4(y0)]-*, since G is abelian, = &(u)~$(y~)[4(y~)]-~, for some u E A (from above), = 4(U)[4(Yo)l-' E A*.
Therefore A * is a subgroup of G. But if a*EA*, then a*=+(x)[&(y,)]-' for some X E A so a*4(yo)E4(A). Since (A1= I&A)I it follows now that &(A) is a coset of A*. This proves Lemma 4.
A.J.W.Hilton, L. Teirlinck
78
Lemma 5 . Let (T, B ) be a proper subsystem of ( S * , a*) of order 3 7 . For i = 0, 1 , 2 let A, = T n Gi. Then, for each i = 0, 1,2, d i ( A )is a coset of a proper subgroup of G.
(T1>7 it followsthat I A I I 2 3 ( i = 0 , 1,2). Let x,#y, betwopointsof A,. Since T is a subsystem, the triple {x,, yl, z,+l}, where +,(x)&(y) = &(z), is in B. so zl+-'is in T. By Lemma 3, T contains the vertical passing through z ~ + and ~, consequently we have that zi E A,. Therefore, by Lemma 4, for i = 0 , 1 , 2 , &i(A) is
prod. Since
a coset of a proper subgroup of G.
Theorem 2. For each integer k 2 1 there exists a non-degenerate plane of order 6k +3.
Proof. The theorem is clearly true if k = 1. Suppose therefore that k 2 2 . For the cyclicgroup G = { 1 , a , a 2 ,..., a Z k } o f o r d e r 2 k + 1 , 1 e tc $ ~ = c # J ,andlet = ~ ~ 42be defined by
42(aJ) = a'''
(06 j s k - 2),
&(a"-') = 1,
d2(ai)= a'
(k =zj s 2k ).
Let ( S * , a*) be the STS derived from G modulo do,dl, 42.Then ( S * , d*)is a non-degenerate plane. For if ( S * , a*) contains a proper subsystem of order 27, then, by applying Lemma 4, G would contain a coset A such that both A and & ( A ) are cosets of the same subgroup of G. But it is easy to see that 42does not in fact transform any proper coset of another coset, a contradiction. This proves Theorem 2. 4. Non-degenerate planes of order 6 k + 1
Let k 3 1 and let G ={ l, a, a', . . . , be a cyclic group of order 2k. Let G, = { a : , a,', a t , . . . ,a 2, k - 1} ( i = 0 .1 ,2 ) be three pairwise disjoint copies of G. Let L ={l,a, a', . . . , a ' . - ' } , R = { a k ,a k + - '.,. . , a 2 k - 1 }and let b, R, be the corresponding subsets of GI( i = 0, 1,2). For each x E G, let x, be the corresponding elements of GI ( I =0, 1,2). Let a be further element (a$G , U G, U G z ) . We construct an STS ( S * * , d * * ) of order 6 k + 1 on the set S**= G,, U G , U G , U {a}.The triples are (i) {xo, x l , x2} (Vx E L ) ; these are called the verticals of the system; (ii) {x,, ( a k x ) , - ,a} . (Vx E L ) ;these are called the obliques of the system; (iii) for i = 0, 1, 2, {x,, y,, z , + ~ }
(Vx, y E G, x f y, xy = z2,z E L ) ; [the subscripts here, and elsewhere, are taken modulo 31
Dimension in Steiner triple sysrems
79
(iv) for i = 0, 1, 2, {xi, yi, (Vx, y E G, x # y, xy = az2,z E R ) . This construction was given by Skolem [9], who showed that (S**, d**) is, in fact, an STS.
Theorem 3. For each k 2 1 there exists a non-degenerate plane of order 6 k + 1.
Proof. W e show that (S**,d**) constructed above is a non-degenerate plane. We consider two cases. Case 1. (S,d**) contains a proper subsystem (T,B) of order 27, and a& T. Since IT1 3 7 it is easy to see that T f l Gi # $3 ( i = 0, 1,2). Moreover, since a $ T, no triple containing a point of T n G i P 1and a point of T n G , is an oblique. Repeating the reasoning of Lemma 3 shows that each point of T is on a vertical, and that the vertical on each point of T is contained within T; consequently T c L0UL,UL2. For each pair xo, yo E T n Go with x , # yo, the third point z1 of the triple containing xo. yo is a point of L,, since T is contained in Lo U L , UL,. Therefore xy = 2’. Then, by Lemma 4, A. = T n Go is a coset of some subgroup of Go. But any such coset must contain at least one point of R. However, since Aoc LO,it follows that (S**, a?**) does not contain any proper subsystem of order 27 which does not contain a.Thus Case 1 does not occur. Case 2. (S**, d**) contains a proper subsystem (T,B )of order 3 7 , and a E T. Since IT\{a}126 it is easy to see that T n G i # g (i=O, 1,2). Let lTnGiI=ti ( i = 0, 1,2). We may suppose that t2 3 max {to, tl}. Suppose that every point x1 E T n G, was on a vertical containing a point of T n G,. Ths would be impossible since it implies that T n R , = @ (by part (i) of the construction), which in turn implies that T n L o = @(by part (ii) of the construction), which finally implies that there are no verticals (!) (by part (i) again). This is a contradiction. We may therefore suppose that there is a point x , E Tn G , which is not o n any vertical containing a point of TnG,. In that case, each of the triples containing x 1 and a point of T f l G , contains also a further point of Tfl G , , with possibly one exception when the further point is not in T n GI, but is instead a. It follows that tl - 1 3t2 - 1. Therefore t , = t2. Since there is a point x, E T n G, which is not on any vertical containing a point of T n G,, it follows that there is a point x2 E T n G , which is not on any vertical containing a point of TnG, (since the verticals contain one point from each of T n Go, T n G , and T n G,). Each of the $ f 2 ( f 2 - 1) pairs of points of T n G2 is in a triple with a point from T n Go, and each point of T n Go is in at most if2triples with pairs of points of T n G,. Therefore there are at least t2 - 1 points in T n Go. Therefore f o = t 2 or t,-1. But since lTl=to+t,+t, is the order of an STS it follows that t, = to. Now let t = t o (= t , = t,).
A.J. W. Hilton. L.Teirlinck
80
The number of triples which contain any given point, and therefore which contain a, is 3t/2. From the construction, the number of triples involving two points from one of T n Go, T n G , , T n G2 and one point from another is 3 x (the number of pairs from a t-set) = 3(5). The total number of triples of d**is f("',"). The number of verticals is, therefore, 3 r + 1 -3( t 3 2 2
-(
)
)--=3t 2
f
2'
The obliques clearly pair off elements of L, with elements of R , + l .It follows that IL,(= (R,I= i t (i = 0. 1, 2), and all elements of L, ( i = 0, 1, 2) are on verticals which are contained in d**. Thus for some X E L it is true that {x,,x,,x2}~JB**. From part (ii) of the construction, it follows that T contains three points a:, a:, a: for some U'E R. From part (iv) of the construction, since { a : , a ; + ' , a { + , } ~ d *(i=O, * 1,2), it follows that a : + ' €T n R, ( i =0, 1,2). Continuing in this way it follows that a:, a:" , . . . , u2k '~T(i=0,1,2),andthenthata:'~T(i=0,1,2).Then,usingpart (ii) of the construction, it follows that a!' E T ( i = 0, 1,2), and then, using part (iv) repeatedly, it follows that a;", . . . , a:-' E T ( i =0, 1,2) also. Thus Ro U R , U R2c T, and, by (ii) again, Go U G, U G2 U {a}E T. But this contradicts the fact that (7'. B) is a proper subsystem. Thus Case 2 does not arise either. Therefore (S**, d**) is a non-degenerate plane of order 6 k + 1, as required. 5. Non-degenerate spaces of dimension 2 3
In view of the results of the last two sections it comes as quite a surprise to realize that there are only two known non-degenerate spaces of dimension 3, namely P(3,2). the projective geometry of dimension 3 over G F (2), which has 15 points and A(3,3), the affine geometry over GF(3), which has 27 points. Similarly there are only two known nondegenerate spaces of dimension d > 3, namely P(d, 2) and A(d, 3). In view of Theorem 11, Section 8, it would be very interesting to have other examples of spaces of dimension d 2 4 such that every subspace of dimension S 3 is non-degenerate. A similar question, and one which may not be too hard to resolve, is t o show that for all sufficiently large n there are STS's of dimension d a 4 and order n which do not contain any proper subspaces of dimension d. 6. An intermediate dimension theorem
P(d, 2) and A(d, 3) provide examples of spaces of dimension d and of orders 2d* - 1 and 3d respectively. But, if d > 3, can they be adapted to provide examples
Dimension in Steiner triple systems
81
of spaces of dimension 3? The theorem in this section, due to Teirlinck [14,15], shows that they can. Its proof is essentially very simple: clearly if, in a space (S, d) of dimension d, one replaces a subspace (T,B) of dimension
of r subspaces of (S, d).Notice that an STS of length 3 is a non-degenerate plane, and that an STS of length 2 is a triple.
Theorem 4. If there i s a space of dimension d 3 3 and order n, then there is a space of dimension e and order n for each e, 2 S e S d. Proof. Since an STS of dimension 2 exists for all orders 3 7 , we may prove the theorem by considering an STS (S, d)of dimension d 3 4 and showing that there is an STS (S, d’) of dimension d - 1 . Let i 3 2 and suppose that replacing a subspace of length S i by another subspace on the same set of points and of any possible length cannot reduce the dimension of S. This is true when i = 2. Let (U,B) be a subspace of length i + 1. Clearly U # S, for otherwise, if S = U and the length of U were S4, then d ( S , d)s 3 , contrary to the hypothesis that d 2 4, whilst if S = U and the length of U were >4, then replacing a subspace of length i by a non-degenerate plane would again give the contradiction that d(S, d)S 3. Let ( U ,V) be another STS and let du= (d\B) UV. We shall show that d(S, d,)3 d - 1. Let u,, . . . , u d - l be d - 1 points of S and let (V, D ) and (V,, 0,)be the subspaces of (S, d) and (S, d,) respectively generated by ul,. . . , Ud-1. Then V # S since d ( S , d ) = d . We shall prove that d ( S , d , ) s d - l by proving that V, # S. Let K , be the set of unordered pairs of points of U (so that ( U ,K,) is the complete graph on the points of V). Let dK= (d\U )U Ku. Let ( V K ,D K ) be the linear subspace of ( S , dK)generated by { u l , . . . ,Ud-l}. [These are the only instances in this section of linear spaces which are not STS’s]. We show that V, # S by considering various cases. Case 1. l U f 7 V K l S l . Then lUnV,l=sl so V , = V # S . Case 2. IU n VKl3 2 . Let xl, x2e U f l V,. Let x3E S be the point such that {xl, x2, x3} is a line of du.Let (U3,B3) denote the subspace of ( U , B) generated by { X l , X2r x31.
AJ. W.Hilton. L. 7eirlinck
89-
Case 2i. U = U 3 .Then vLJ
[Ul,.
..
7
Ud-11
u]du= [ u l , . . .
I
u d - 1 , u ] d = [ U l , .. *
1
Ud-1,
x31d.
But the subspace of ( S , d) of which [ u , , . . . , q - 1 , x,] is the ground-set has Consequently V,# S. dimension S d - 1 and thus is a proper subspace of (S, a). Case 2ii. U # U,. Then U,C U. Case 2iia. B 3 contains exactly one line of B, and ( U ? ,B,) is a maximal proper subspace of ( U , €3). Let X,E U\{x,, x2, x3). Then
so V" # s. Case 2iib. Either B , contains more than one line of B or (U3,B3) is not a maximal proper subspace of ( V .B ) . Let (T,9) be a maximal proper subspace ( U , B ) with x , . x?, x , E T , and let ( T . G ) be a non-degenerate plane with {x,. x?. x 3 } e G. Let d., = ( d \ 9 ) U (3. T h e n d ( S , & & , . ) a dsince the length of (T,.9) is Gi and, by assumption, the dimension of ( S , d ) cannot be reduced by replacing a subspace of length Gi by any other subspace. Let (V.,., DT)be t h e subspace o f ( S . d T )generated by { u , . . . . vd ,}. Consider the following two subcases of Case 2iib. Case 2iib 1. V , f' U ={x,, x2. x,}. Then (V,, DT) is a proper subspace of (S,.dT).Since { X , , X ~ , X , } E S ~ , it follows that ( V T , D T ) = ( V u , D L JSO) that (V,, D,) is a proper subspace of ( S , d,),and so V , # S . Case 2iib 2 . V, f?U # { x , , x2, xD}. For u E U let the subspace generated by {v,, . . . , v d - , , u ) in ( S , d T ) be denoted by (W,,, 8,").Then d(W,,, % T u ) s d - l . Case 2iib 2i. V , n U .3 T. Let y E U\ T. Since the dimension of (S,d-,.is) at least d, it follows that ( W,,, S T , ) #is, d T ) .Since ( T ,9)was a maximal proper subspace o f (U. R ) it follows that U c W,,, so W , , = l u , . u 2 . . . . , cd. ,,UIdL,.Thus
.
v,
~
5 s.
wTy
Case 2iib 2ii. V, n U # T. Since x,, X ~ V E , c V, and {x,, x2, x3} is a line of G c d, it follows that { x , , x 2 , x , } c V, n T. In fact V, n T = {x,, x2, x3}, for otherwise. since (T. 3 ) is a non-degenerate plane, it would follow that T = V, n Yc V, fl U , a contradiction. Since V, f' U # { x , , x2, x,} there is a point x E ( U \ T) f l Vp Let z E T\{x,, x 2 , x,}. Then x E W,, and T c WTz SO it follows that U c WT,. Thus W,, is the ground-set of Some subspace of (S, Sa,). Since d ( S ,d o )= d, W, # S. But V , c W,,, SO V, # S, as required. It now follows that d ( S , &,)a d - 1 . Now, given (S, 4)of dimension d, if we replace a maximal proper subspace by a non-degenerate plane, we can construct an STS of dimension S 3 . Thus, for some least value & a 2 of i it is true that replacing a subspace of length i by any other subspace cannot reduce the dimension, but replacing a subspace of length i + 1 by another subspace can reduce the dimension. Then, in that case, the dimension will be d - 1. The theorem now follows.
Dimension in Sreiner triple systems
7. The existence of STS’s of dimension d
83
23
In this section we show that STS’s of dimension d 5 3 exist for all sufficiently large orders n. The key to this is the singular direct product construction for STS’s. First we give some preliminary definitions. A latin square L = ( l i i : 1 S i S n , l S j s n ) of order n on the set X = {x,, . . . ,a}, where x i # x i if i#j, is an n x n matrix such that each element of X occurs exactly once in each row and column. A tricouer of order n on X is an n 2 x 3 matrix M =(pi : 1 S i S n2, 1S j s 3) such that each ordered pair (xv, x ~ ) , where 1 svsn and l s p s n , occurs exactly once as a row in each n 2 x 2 submatrix of M. It is well-known that a tricover corresponds to a latin square, since corresponding to L we have the matrix M given by x,
if j = 1,
lye, if j = 3, where i = ( y - 1)n + S and 1Si3S n, which is a tricover of X . Clearly a tricover exists for each order n 3 1. We now describe the singular product construction. Let ( S ’ , d ’ ) , ( S , d ) and ( T ,B ) be STS’s of orders u3, u2 and u1 respectively, and let (S‘, d’) be a subsystem of (S,d).Let R be the set of rows of a tricover of S\S’. Let (T, B, 6) denote the STS (T,B) with each triple of B assigned an ordering. Let (S‘U{(S\S’)x T}, ( B , 6 ) 0 ( d , d ‘ ,R ) denote the STS on S ‘ U { ( S \ S ’ ) X T } of order u , ( u 2 - u 3 ) + u 3 whose triples are (i) {a,?aj?ak} for {&, a,, a k ) E d‘, (ii) {a,, ( 4 ,b,), ( a k , b,)} for all {a,, a,, ak}E d with ai E s’, 4,a k E s\s’ and for all 6, E T, (iii) {(U,, b,), (aJ,b,), (ak, b,)} for all {U,, U,, ak}E d with U,, a,, uk E s\s’ and for all b, E T, (iv) {(U,, br), (aJ,bs),(ak, 4))for all {(U,, 4. ak)ER, and for all (br, b,, b t ) E ( B ,6 ) . Then (S’U{(S\S‘)x T } ; (B, 6 ) 0(d, d‘, R ) ) is the singular direct product of ( S , d),(S’,d’), (T, B) with respect to R and 6. This construction was essentially due to Moore [8]. Notice that we can be sure that the singular direct product contains (T, B) as a subsystem by requiring that (s, s, s) E R for some s E S \ S’.
Theorem 5. Ler ISI>JS’I. If (T,B ) has dimension d , then
has dimension at least d. Proof. Let e l , . . . ,c, be any d points in S’U((S\S’)x T}. We may suppose that for some e, with O S e S d , {cl , . . . , c , } c S ’ and { c e + ,] . . . , cd}c_(S\S‘)xT.
84
A.I. W. Hilton. L. Teirlinck
Suppose c,=(a,,b,), where a,eS\S', b , c T ( e + l a i C d ) . Then {be+,,..., bd} generates a proper subsystem ( T ' , B ' ) of (T,B), and so {c,,. . . , c J generO')U(d,d', R)).(Here ates a subsystem contained in (S'U{(S\S')x T'}, (B', (T', B', 0') is the restriction of (T. €3.0) to T'.) This proves Theorem 5. We have immediately from Theorems 4 and 5: Tbeorem 6. Ler ( T ,B ) be an S T S of dimension d a 2 and order u , . Let ( S , d)be an STS of order u2 with a proper subsysfem ( S ' d ' ) of order u,(
Theorem 7. If A and p are admissible integers wirh I.L 3 2A + I, then there is an STS of order p which contains a subsystem of order A. The main result is the following theorem.
Tbeorem 8. I f there is an STS of dimension d 3 (mod 6) and
23
and order u, and if n = 1 or
4v2+2u - 3 for u = 1 (mod 6 ) , 4 v ' - 2 u - 3 for u = 3 (mod 6),
then there is un STS of dimension d and order n.
Proof. If there is an STS of dimension d and order u, and if A and p are admissible integers with p 2 2h + 1. then, by Theorem 6 and 7, there is an STS of dimension d and order A + u ( p - A ) . If (1) holds it may easily be verified that appropriate values of A and p such that n = A + u ( p - A ) are given in Table 1. There q, r and i are integers satisfying n = 6qu + 6r + i, i = 1 or 3. 1 c 6 r + i =s 6t. - 3. The largest values of A in Table 1 which give rise t o the numbers in (1) occur when A={
4 ~ - 3 = 6 r - 2 ~ + 3 if u = l ( m o d 6 ) , 6 u - 3 = 6r+3 if u = 3 (mod 6).
This proves 'Theorem 8. Since P(d, 2) is an STS of order u = 2d ' theorem.
-
1 we immediately have the following
Dimension in Steiner triple systems
85
Table 1. Range of r
i
u
Value of
Value of A
1 lS6r+lS4u-3 1 4~+3S6r+lS6u-5
6q+6r+l 6r+l 6 q + 6 r - 4 ~ + 5 6r-4u+l
3 3
3<6r+3S211-5 2u + 1s 6 r + 3 < 6 u - 3
6q+6r+3 6r+3 6q +6r-2u + 5 6r-2u + 3
1 1
ls6r+1<2u-5 2u + 1 s 6 r + 1~ 6 - 5u
6q+6r+l 6r+l 6q +6r -2u + 3 6r -2u
+1
3 3
3S6r+3S4u-3 4u + 3 < 6 r + 3 S 6 u - 3
6q+6r+3 6r+3 6q +6r -40 + 7 6r -4u
+3
-1
r 3
Theorem 9. If d 3 3, n is admissible and 22d+4 - 3 2d+2- 1 when d is even, > 22d+4-5* 2d+2+3 when d is odd,
{
-
then there is an STS of order n and dimension d .
8. Further related notions A projective hyperplane of an STS (S, d)is a proper subspace ( H , B) of ( S , d) such that any line of d has at least one point in H. Teirlinck [14-161 proved that a hyperplane ( H , B) of (S, d)is projective if and only if IHI = +(IS1- 1).H e then went on to associate canonically a projective space with each STS, by proving the following result.
Theorem 10. If (V, %’) is the intersection of all projective hyperplanes of an STS ( S , d),then the lattice of all subspaces of ( S , d)containing V, ordered by inclusion, is a projective space over GF (2). A similar related result was proved by Coupland [l]. The projective dimension d,(S. d)of ( S , d)is the dimension of the projective space. Clearly &(S, d)S d ( S , a). Teirlinck also showed that there are at least 511 non-isomorphic STS’s of order 31 with 4 2 3 . Teirlinck [14-16] also analogously defined and studied the affine dimension d*(S, d)of ( S , 4. We now mention some relationships between linear spaces in general and matroids. A matroid will be taken to be a pair ( X ,M), where X is the set of points and M is the set of independent sets. The rank function of ( X ,M) will be denoted by p. A matroid is projective if IAlc2, A c X + p ( A ) = A and is plane if p ( X ) = 3 . If ( X ,M ) is a matroid of rank k 2 1, then an erection of ( X , M) is a matroid of rank
86
A.J.W. Hilton. L. Teirlinck
k + 1 which has the same subspaces of rank = ~-k1 as has (X, M). If there is an erection, then there is a unique erection, called the free erection, which is minimal in the sense that if H is any hyperplane of the free erection of (X, M), and (X, M') is any other erection of ( X , M), then ( X , M') contains a hyperplane K such that
H c K. Firstly, if ( X , B) is a linear space of dimension 2 4 then, by forming the restricted rank function r3 defined, for each Y E X , by
or equivalently the "cut-off' family 9 ( 3 ) of /-independent sets, where A ~ 9 ( 3 )if and only if
A € 9 and (A1s3,
then ( X , $ ( 3 ) ) is a plane projective matroid. Conversely, givcn any projective matroid we can take the maximal dependent sets of rank 2 as the lines, and thereby define a linear space. Looked at this way, the dimension function becomes a natural generalization of the rank function of plane projective matroids. Now consider the question of when the set 9 of I-independent sets of (X, B ) is the set of independent sets of a matroid. Let
9 ( k )= { A: A€ 9 ,IAI6 k } . In [6] is proved the following theorem.
Theorem 11. Let 1 =sk S d ( X ) + 1. Then ( X , 9 ( k ) ) is a matroid i f and only i f : (i) to each linear subspace (Y, %) with d ( Y , % )ksthere is exactly one linear subspace ( Z , D ) with d ( Y , %) = d ( Z , D ) , ( Y, %) c (2,D ) and such that each linear subspace ( W, 8)containing ( Z , D ) has dimension greater than d ( Y, %), und (ii) no linear subspace of ( X , B) of dimension < k - 1 is very degenerate. This has the following corollary. Corollary. I f ( X ,$ ( k ) ) is a matroid, then d
+ 1 is its rank function.
We also have the following result about the free erection of ( X , $ ( k ) )
Theorem 12. Let k a 3 . I f ( X , $ ( k + l ) ) is a matroid of rank k + l , then t X . . 9 ( k + l ) ) is the free erecrion of ( X , $ ( k ) ) . References [ 1 1 J . Coupland, On the construction of certain Steiner systems, Ph.D. Thesis, University College of Wales. May. 1975. [21 J. Doyen, Sur la structure de certains systkmes triples de Steiner, Math. Zeit. 111 (1969) '89-300.
Dimension in Sreiner triple systems
87
131 J. Doyen, Systtmes triples de Steiner non engendrts par tous leurs triangles, Math. Zeit. 118 (1970) 197-206. [4] J. Doyen and R. Wilson, Embeddings of Steiner triple systems, Discrete Math. 5 (1973) 229-39. [5] A.J.W. Hilton, On the Szamkolowin-Doyen classificationof Steiner triple systems, Proc.London Math. Soc. 34 (3) (1977) 102-116. [6] A.J.W. Hilton, Dimension in linear spares, Proc, 1976 Conference on Graph Theory and Combinatorics, Orsay, CRNS publication No. 260, 233-236. [7] M. Las Vergnas, On certain constructions for matroids, Proc. 5th British Combinatorial Conf. (1975) 395-404. [8] E.H. Moore, Concerning triple systems, Math. Ann. 43 (1893) 271-85. [9] Th. Skolem, Some remarks on the triple systems of Steiner, Math. Scandinav. 6 (1958) 273-280. [lo] L. Szamkolowin, On the problem of existence of finite regular planes, Colloq. Math. 9 (1962) 245-50. [ l l ] L. Szamkolowin, Remarks on finite regular planes, Colloq. Math 10 (1963) 31-37. [ 121 L. Szamkolowin, Sur une classificationde triplets de Steiner, Rend Accad. Naz. Lincei 36 (1964) 125-1 28. [13] L. Szamkolowicz, Alcuni problemi della teoria dei sistemi di Steiner, Rend. Mat. e Appl. 24 (1965) 348-359. [14] L. Teirlinck, On Steiner spaces, J. Combinatorial Theory (A) 26 (1979) 103-114. [ 151 L. Teirlinck, Combinatorial structures, Doctoral dissertation, Vnje Universiteit Brussel, 1976. [ 161 L. Teirlinck, On projective and f i n e hyperplanes (to appear).
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Annals of Discrete Mathematics 7 (1980) 89-94 @ North-Holland Publishing Company
PERPENDICULAR ARRAYS OF TRIPLE SYSTEMS E. MENDELSOHN Department of Mathematics, University of Toronto, Toronto, Ontario, Canada, M5A lA7 Let V be a d-tuple, V=(u,. ul, u 2 . . . ud-&, then if l c { O , 1,. . . ,k}, let V, = UicIV,. A PATS (n,d) is a collection V of in(n-1) vecton V={u,,u,,u, . . .u,-,} such that if 111=3, then every pair {u, u } of V is contained in exactly one V,. Some necessary condition on the existence of PATS (n, d ) are given and some constructions are shown.
Introduction In [l] Bose introduced the concept of an orthogonal array and showed it was co-extensive with the existence of mutually orthogonal latin squares. Attempts to extend the concept of orthogonal to a more general type of designs leads to the concept of perpendicular quasigroup [5], orthogonal triple systems [6], and perpendicular arrays of t-designs [3]. For the sake of comparison we repeat the definition here.
Definition 1. A pair of commutative quasigroups (A, .) and (A, *) are called perpendicular iff given any unordered pair { x , y } , there exists u , b such that { U * 6, u * b } = { x , y } . Definition 2. A Steiner triple system (S, 9)is a set S together with a collection of its three-subsets 9 so that every pair occurs is in exactly one member of 3.(S, 9) and (S, 9')are said to be orthogonal if 9 n9'= fl and if xyz and uuz E 9, there is no z' such that xyz' and U U Z ' E ~ ' . Definition 3. A t-(u, k, A, b ) design (V, 9)is a collection 9 of b subsets of size k of V, such that every subset of size t of V occurs in A subsets of 9.A perpendicular array of t designs is defined as follows: PA ( t , u, k, A, b, d ) , (V, €3) is a set V together with a subset 93 c Vd with the following property: Let I c (0,1,2, * * .d - l},define for u E 9, V,= U ui. For every I, 111 = k define 9r= {UI,u E B} then (V, $33,) is a t-(u, k, A, b ) design. A PATS (n,k ) is simply the case of P A (2, u, 3, 1,i u ( u - l),d) design.
Some necessary conditions We now give some necessary conditions on the existence of PATS (n, k). 89
E. Mendelsohn
VO
Lemma 1. d = 3 . 4 . Proof. d # 1 , d # 2 trivially and d = 3 is just a Steiner triple system. If d 3 5, let ( a , b. c, d, e . * . ) E V. Now {a, b} must occur in IJ 9 1 2 4 . So there must be a vector u = (x, y, a, b * . .) or (x, y, b, a . . .). Similarly for B i z 4 we must have u = {x, y, z, a, b } or {x, y, z, b, a} but then S , has two occurrences of a, b. From now on we shall mean by a PATS ( u ) a PATS (u, 4). Corollary 1. In a PATS ( u ) each pair { a , b} occurs in two 4 tuples and in complementary places. Corollary 2. A PATS (v) is naturally a 2-(u, 4, 2) design. Lemma 2. If PATS ( u ) exists u = 1 mod 6.
Proof. As (V, a,,,,)is an STS, we must have u = 1 , 3 mod 6. Let
the number of times a occurs in the ith place of a 4-tuple in a PATS ( u ) . We then get
Thus . r p = r f= A( u - 1). Thus u
r f be
l(6).
Lemma 3. If a PATS ( u ) , u, contains a subsystem, a PATS (w),then u 3 3 w + 4 .
Proof. Let us denote the elements of W by capital letters and those of V- W by small letters. Let V- W = 6t. We then have five different kinds of 4-tuples and the number of each is defined by the following table: A B C D iw(W-1) no A b c d n1 a B c d
a b c d n2 a b c D n3 a b c d n,.
As blocks have the complementing pair property (Corollary 2) we deduce n o = n , = n, = n3. Let us count capital -small pairs in B,,,. We obtain 3n, = 6 1 . $W or no= t . W. By counting small letter pairs we obtain 3 n 0 + 3 n 3 + 3 n 4 = (?), i.e. n 4 = t . [ 6 t - 1 - 2 W ] . Thus, 6 t 3 2 W + 1 . As 2 W + l = 3 ( 6 ) we must have 6 t * 2 W + 4 or V * 3 W + 4 . The fact that no simple recursive construction has been found of the form u -+ 3u + 4 can be attributed to the fact that the arithmetical constraints are not exact. Compare the case of Steiner triple systems t o those of Room squares [4]. In t h e latter t h e constraints also must be adjusted for congruence class and no minimal construction has yet been found.
Perpendicular arrays of triple systems
91
Asymptotic results
We shall show in this section:
Theorem 1. For N >No, N = 1mod (6) there is a PATS (N). As a first approximation to this theorem, we shall show:
Theorem 2. There exist infinitely many N such that a PATS ( N ) exists. The proof of this depends on the following construction in a finite field.
Lemma 4. Let K be a field IKI = qu, q" = 1mod (6). Let 5 be a primitive root and let T , be the subgroup of K* generated by e3. Let TA= T t , TB= Tt2. Let P c T,, P ={l,t3,5". t9* * - r3'},s < t where q" = 6t + 1. Let there exist a, b E K* such that Table 1 is satisfied: Table 1
a b a-b a-1 b-1
E
#
TA
*1
T,
*1
T, TR TA
+1 *b *a
Then the following array is a PATS ( q u ) . {(ox+ y , 1x + y , ax + y , bx + y ) I x E P ; y E K}. Proof. The sets (0, x, ax},(0, x, bx},{x, ax, bx}, x E P contain all distinct differences exactly once and so we obtain a cyclic STS on each column [2]. We now apply the theorem of Wilson [7] which shows that the conditions of Table 1 are satisfied infinitely often to obtain Theorem 2. We now check some small values of prime powers to see if this result holds early.
Theorem 3. a cyclic PATS ( q u ) exists according to the following table for 7 s p' 67: n = 7 : None; (No PATS exist at all). n = 13: None; (No PATS exist at all).
S
92
E. MendeLFohn
Definition 4. A pairwise balanced design with A = 1, and block sizes {ki}i,r on u vertices is a collection of subsets called blocks of (0 * * u - 1) of sizes taken from {/c,},er so that every subset of size two occurs as a subset of exactly one block. We denote such a design by (u, k,;1).
Lemma 5. I f a (u, k,; 1) exists and PATS (k,)exists for all k,, then a PATS ( u ) exists.
Proof. Let 9 be the set of blocks of a (u, k,, 1) design. Let B E ~ , =B (ui, u2, . . . , u,J. There is a PATS (k,),A, which can be taken to be on the symbols u , . . uk. Then A = { A B1 B € 9 )is a PATS ( u ) . For if one chooses a pair of elements of u and three places of A, then that pair determines a block B in the design and in the AR that pair occurs exactly once among three places chosen.
Theorem 4. If N = 1(6), N > N o , then there exists a PATS (N).
Proof. By Lemma 5 we need only show the existence of No for the class pairwise balanced designs whose block sizes are the sizes of PATS(n). There is a PATS (19) and PATS (25) by Theorem 3. Now as both (18,24), and ( 1 9 - 18.
Perpendicular arrays of triple systems
93
25 - 2 4 ) = 6 , by the Wilson’s Theorem [8] the desired No exists for PBD’s with block sizes 19 and 25. We remark that this also works for 31 and 49.
Theorem 5. If N = =1(6),N >M,, then there exists at least two non-isomorphic PATS (N). Proof. By using Wilson’s theorem on block sizes 19 and 25 and on 61 and 67 we arrive at an Mo for which there are two daerent constructions for all N = 1(6), N aM0. We notice that in the first of these there is a pair of elements xo, yo such that the smallest subsystem containing them is of size 25 if indeed there is a block of size 25 in the PBD. But every two elements in the second construction generate a subsystem of size at most 31 or 49. But 3 * 25 + 4 >49 so they cannot be all isomorphic. If the PBD in the first case uses only blocks of size 19, only one has also 3 - 19+4>49. Thus there are at least two non-isomorphic systems. Product constructions
In this section we wish to give an explicit construction which will prove the following result.
Theorem 6. If a PATS ( u ) exists and a PATS ( u ) exists, then a PATS (uu) exists. Proof. For a set u define A,, = {u, u, u2, u3}.Furthermore if A and B are subsets of V4 define A @ B to be bo), (a17 bi)?( 0 2 ,
{((a09
b3) I
(a07 a19
a23
as) E A, (bo, bi, bi, b3) E B}.
Let us have a PATS ( u ) on a set (V, B) and PATS ( u ) on a set (U, A). Let (V, C) be an orthogonal array obtained from two mutually orthogonal latin squares of side I VI. On U x V form the following 4-tuples, A @ C U AA @B. Note this is a set of ~[u(u-l)u2+[u(u-l)]u]=~[uu(uu-1)], 4-tuples. Thus we need only show, given any pair and three places, we can find that pair once among the three places. Let (uo, uo) and (ul,ui) be given uof ul. Then in A we find uo, u1 as a pair once amongst the three places given. Now in C, { u o , u l } occur between every pair of places. Thus in A@C, (u,, uo), (ul,ul) occur once. If uo= ul,then in A, @B we find (uo, u,), (uo, ul) appropriately. We note that these methods leave large gaps uncovered for example between 67 and 247 we cannot deal with 85, 91, 115, 133, 145, 175, 187, 205, 217 and 235. We close with the following conjectures (1) If n 3 6 7 and N = l(6) a PATS (N) exists. (2) If I ( n ) is the number of non-isomorphic PATS (6n + 1), then I ( n )-+ co as n +a. (3) Singular direct product construction can be modified to work for PATS.
94
E.Mendelsohn
References [l] R.C. Bose. A note o n orthogonal arrays, Ann. Math. Stat. 21 (1950) 304-5.
[2] L. HetTter, Ueber Nachbarconfigurationen, Tripelsysteme und metacyklische Gmppen, Deutsche Mathem. Vereinig. Jahresber. 5 (1896) 6 7 4 8 . [3] E. Mendelsohn, Perpendicular arrays of regular tournaments (to appear). [4] R.C. Mullin and R.G. Stanton, Techniques for Room squares, Proc. Louisiana Conf. on Combinatorics, Graph Theory and Computing, Louisiana State Univ.. Baton Rouge, La (1970) 445-464. M R 4 2 (1971) #1679. [5] C.D. O’Shaughnessy, On Room squares of order 6m+2. J. Combinatorial Theory (A)13 (1972) 306-314. MR46 (1973) #1619. [6] T.G., Room, A new type of magic square, Math. Gaz. 39 (1955) 307. [7] R.M. Wilson,Cyclotomy and difference families in elementary abelian groups, J. Number Theory 4 (1972) 1 7 4 7 . [8] R.M. Wilson, An existence theory for painvise balanced designs 11. The structure of PBD-closed sets and the the existence conjectures. J. Combinatonal theory (A)13 (1972) 246-273. [9] R.M. Wilson, Private communication.
Annals of Discrete Mathematics 7 (1980) 95-104 @ North-Holland’ Publishing Company
STEINER SYSTEMS AND ROOM SQUARES R.C. MULLIN Department of Combinatorics and Optimization, Uniuersity of Waterloo, Waterloo, Ontario, Canada
and S.A. VANSTONE St. Jerome’s College, Uniuersity of Waterloo, Waterloo, Ontario, Canada In 1968, O’Shaugnessy gave a construction for Room squares in terms of Steiner triple systems. Since then, various authors have developed this theme and its generalizations. It is interesting to note that in 1863 Cayley had also established a relationship between Steiner triple systems and Room squares which is different from that of O’Shaugnessy. This paper surveys various results connected with the above.
1. Introduction In [14], T.G. Room defined what is currently called a Room square. It was pointed out later by N.S. Mendelsohn and B. Wolke that such arrays had been discussed by Howell in connection with duplicate bridge tournaments (see [ 11, for example). The authors are indebted to A. Rosa for acquainting them with what is, to our knowledge, the earliest occurrence (due to Cayley [4]) of a Room square in the literature. A survey of these configurations, their generalizations, and their relationship to Steiner systems is the subject of this article. Without regard for chronological order, we begin with a generaIization of Room squares, which will be specialized in later sections.
2. LORS and perpendicular Steiner systems A labelled OR-square (for orthogonal square) of degree t index m, strength A and order n (briefly LORS ( f , m, A, n) is a square array where each cell is either empty or contains a 1-subset of an n-set S, and each row and column of the array is labelled by an m-subset of S called label, such that (i) for every m-subset of S there is exactly one row and column labelled by it, (ii) each element of S, except the elements of the label, occur in every row and column exactly A times, and (iii) every t-subset of S occurs in exactly one cell of the array. A LORS ( t , m, A, n) is normalized if for every i, the ith row and ith column are labelled by the same m-subset of S, and the diagonal cells are empty. 95
R.C. Mullin. S.A. Vanstone
96
It is assumed that the reader of this volume is familiar with Steiner systems. (We use the parametric notation S(t, k, u ) and require that D > k.) Two Steiner systems (V, F,) and (V, F2) are said to be perpendicular if (i) they are disjoint, i.e., F, n F, = $4, and (ii) whenever B and B’ are two blocks of F, which intersect in a (k - t)-subset P, and R, R’ are such that (B \ P) U R E F, and (B’\P) U R’ E F2, then R # R’. The relationship between perpendicular Steiner systems and LORS is given in [18] by the following theorem. Theorem 2.1. Let t < k S 2 t - 1. lf there exists a pair of perpendicular Steiner systems S ( t, k, D), rhen there exisrs a LORS (1, k - t, A, u ) with
A=(~-k+t--l)/( r-1 21-k-1 2r-k-1
)
The following examples of perpendicular LORS (2, 1,3,8) are given in [18].
S(3,4,8)’s and
associated
V={O, 1.2,3,4,5,6,7), B,:012 0 1 3 0 1 5 0 2 3 0 2 6 0 3 4 0 4 5 0
467 157 126 237 134 245 356
4 7 6 5 7 6 7 1
1 1 1 1 2 2 3
2 2 3 4 3 4 5
3 5 4 6 4 5 6
2
235 346 037 456 247 567 057 135 367 016 034 147 026 045
B,:01 0 1 0 1 0 2 0 2 0 3 0 4
6 7 5 7 7 5 7 3
4
5
26 3 4 5 7 3 7 4 5 5 6 6 7
1 1 1 1 2 2 3
6
457 156 267 137 056 236 024 257 014 067 347 035 025 017 145 167 036 012 246 127 047 027 357 123 125 013 146 234
2 2 3 4 3 5 4
3 4 6 5 4 6 5
5 7 7 6 6 7 7
7 124 345 136 046 256 023 015
The following necessary condition for the existence of perpendicular Steiner systems is given there. Theorem 2.2. If there exists a pair of perpendicular Steiner systems S ( f, k, u ) , then
Steiner systems and Room squares
97
As pointed out in [18] an immediate corollary of the above is that there exists a pair of perpendicular S(3,4, u)'s if and only if u = 8. A Steiner k-tuple system is an S(k - 1, k, u ) . A further corollary to the above theorem is the following.
Corollary 2.3. There are no perpendicular pairs of Steiner k-tuple systems for k 3 5.
Proof. First note that in a S(k- 1, k, u ) , the relation u 2 2 k must hold. Indeed, the complement of such a design is a (k - 1)-design in which A k - l = (x-"&:l)/k. Since u > k , we have Ak-121. Hence u - 2 k + 1 2 1 , i.e. u 2 2 k . Now for u z 8 , if 4 s k k [ ~ u ] + l then .
thus for u a 8 , 4 s k S Liul+ 1. we have ( k U , ) ~u ( u - 1)with equality if and only if u = 8 and k = 4. By Theorem 2.2, it follows that in a pair of perpendicular Steiner k-tuple systems with k 3 5, we must have u ~ 7which , contradicts the fact that u 2 2 k . 0 Thus the only possible perpendicular pair of Steiner k-tuple systems apart from those with k = 4, u = 8 are Steiner triple systems. These are discussed in greater detail later. Probably the most interesting pair of perpendicular Steiner systems apart from those corresponding to Room squares (discussed later) are those arising from the Witt designs S(5,8,24). Essentially Kramer [6] has used such designs to create a perpendicular pair of S(4,7,23)'s.
3. Generalized Room squares A generalized Room square of degree k and order n + 1 (written GRS (t, n + 1)) is an ( k l lx)( k y l ) array such that (i) each cell of the array is either empty or contains a k-subset of an ( n + 1)-set X, (ii) every element of X occurs in each row (and each column) of the array in exactly one cell, and (iii) each k-subset of X appears in exactly one cell of the array. In particular, a Room square of side n is a GRS (2, n + 1).Clearly by Theorem 2.1 a GRS (t, n + 1) can be obtained from a perpendicular pair of S(t, 21 - 1, n)'s and a Room square can be obtained from a perpendicular pair of S ( 2 , 3 , n)'s, that is, a perpendicular pair of triple systems FTS (n). (The trivial case of a GRS (k, k) will not be considered subsequently). We consider the cases of t > 2 and t = 2 separately. (In passing we note that a GRS (1, n + 1) is a latin square.) We first consider the case t > 2. For t a 4 there
98
R.C. Mullin. S.A. Vanslone
are only 2 GRS known, namely the G R S (4, 12) found by Rosa and Blake [see 151 (which cannot come from a perpendicular pair of Steiner systems), and the GRS (4,24) found by Kramer [6] and Blake and Stiffler 131. Although their method does not explicitly employ perpendicular Steiner systems, it can be shown that their method produces G R S isomorphic to those arising from perpendicular pairs of S ( 4 , 7 , 2 3 ) . In fact, as mentioned in [ 181, Kramer's result implies the existence of a set of 9 Steiner systems S ( 4 , 7 , 2 3 ) such that any pair of distinct members of the set are perpendicular. For t = 3, there are infinitely many orders for which G R S (3, n) are known to exist. The first such family was found by Stiffler and Blake, who use a recursive method and initial configurations to show the existence of GRS (3,7" + 2 ) and GRS (3, 16" + 2) for a = 1 , 2 , 3 , . . . Other examples of GRS (3, 12) have been constructed by Blake and Stiffler [3] who use the affine group over G F (4) and a hand algorithm to construct G R S (3, n) for n - 1 a prime and for n satisfying 6 < n < S O . Further, Rosa [ 151 shows that the existence of GR S (3, n ) with n even implies the existence of G R S ( 3 , 3 n ) and also shows the existence of GR S (3, 15) and independent from the above citation, G R S (3,9). Unfortunately the above does not shed much light o n the existence of perpendicular pairs of S(3,5, u ) . The authors are unaware of the existence of any such systems.
4. Room sqnares and perpendicular triple systems It is known [ 1 11 that a necessary and sufficient condition for the existence of a R S ( n ) is that n be an odd integer 2 7 . Unfortunately no such conditions are known for the existence of PTS(n). We list below results on the existence of PTS (n).Clearly a necessary condition is that u be congruent to 1 or 3 mod 6 and that u be 3 7 .
Theorem 4.1 (Mullin and Nemeth [9]). There does not exist a PTS(9). Theorem 4.2 (Mullin and Nemeth [lo]). If u is a prime power and u = 1 mod 6, then there exists a PTS ( u ) . Theorem 4.3 (Rosa [17]). There exists a PTS(27). The above constructions are direct. Some recursive constructions are given below. It is assumed that the reader is familiar with pairwise orthogonal latin squares. orthogonal arrays, pairwise balanced designs (PBDs) and PBD closure.
Theorem 4.4 (Horton [5]). If there exisrs a PTS(u,) and a PTS ( u z ) which contains a subsystem PTS ( u 3 ) ,and i f there exists a set of u1- 2 pairwise orthogonal latin squares of order u2- u3, then there exists a PTS (u1(u2- u3)+ U J .
Steiner systems and Room squares
99
Theorem 4.5 (Lawless [7]). Let P = {u :3 PTS (u)} . Then P is PBD closed. (That is, if there exists a PBD (u,k ) and if for each k E K there exists a PTS (k), then there exists a PTS (u)).
In view of Wilson’s theorem on PBD closure [20], Theorems 4.2, 4.3 and 4.5 imply that there exists an integer uo such that if u > uo and if u = 1 or 3 mod 6, then u E P, where henceforth we assume P = { u :3 PTS ( u ) } as in Theorem 4.5. Despite the power of these methods, 27 and 189 are the only integers congruent to 3 m o d 6 less than 350 for which PTS exist, to the authors’ knowledge. Another aspect of PTS’s is treated by O’Shaugnessy [12], who shows that the PTS’s of the construction of Theorem 4.2 can be used to produce many others of the same order.
5. Cayley’s construction
As mentioned, to the best of our knowledge, the earliest occurrence of a Room square in the literature is in a paper [4] by A. Cayley in 1863. Hence, a more appropriate term might be “Cayley square”. This, of course, will not be used since the present terminology seems well established. The following RS (7) and labelling is that which appears in Cayley’s paper.
a
b
62 13 47 58 12 36
78 45
c
84 57 16 23
d
e
f
g
35
17
82 15
64 37
86
42 38
14
56 27
25 67 34
18
The purpose of constructing such an array is to display a solution to the Kirkman school-girl problem which contains a subset of 8 elements, no three occurring in a common triple. The elements of the RS (7) form a subset of size 8 and the solution is as follows.
R.C. Mullin. S.A. Vansrone
100
This construction for Kirkman triple systems from Room squares can be generalized whenever the side of the Room square is a prime power congruent to 1 modulo 6. Before showing this, we require several definitions and results. We assume standard results on balanced incomplete block designs (BIBD's). Let G be a finite additive abelian group of order 2n + 1. A strong starfer in G is a partition of the non-zero elements of g into pairs {x,, yl}, . . . , {x,,, y,,} such that (1)
6{*)XI - Y,)}
,=I
(2)
x,+y,#O,
(3)
x,+y,#x,+y,,
= G\{O).
l ~ i ~ n . for all i,; (if;).
+ y,) is called the adder element for the pair {x,, y,}.) It is well known that the existence of a strong starter implies the existence of a Room square of side 2n + 1. Let u be a prime power such that u = 2"(6m) + 1 where a is a non-negative integer and m is odd. Consider the finite field GF(u). Let x be a primitive element in this field. It was shown by Mullin and Nemeth [ll] that if d = 2", then the set of pairs {{X2~dX i ~( 2 ~ + I ) d +s~ ;) -=d--l,O=si=s3rn-l} :~ (--(x,
is a strong starter in the additive group of GF ( u ) . They also showed that if a is an adder element then -a is not. In this case, the adder elements are said to form a skew adder. Let A be the set of adder elements. Let e = 2"+' and H ebe the subgroup of GF* ( u ) consisting of the eth powers of x. Clearly, IH'( = 3m. Since 2e C u - 1, - 1 $ H e . It is well known that the cosets of H' form the base blocks for a (c, m, rn - 1)-BIBD. Now the 3mth powers of x form a subgroup K and - 1 E K since 6 m 1 ( u - 1). Thus if H: is the coset of H' which contains a, then H',# H:. But the differences in H', are the same as those in HT, and hence, if S is a system of distinct representatives for the cosets of the factor group K / { 1, -I}, then { H : : s E S} is a set of base blocks for a (u, rn, i(m - 1)BIBD. Now A consists of a union of cosets of H' such that if the elements of H', are in A. the elements of H',are not (skew adder property). Let S' be a system of distinct representatives for these cosets. Then, S = K\ S' is a system of distinct representatives of the cosets of K/{l, -I}. Since 3 I3m, the subgroup of order 3 induces a partitioning of the cosets H,, s E S, into triples such that the set of all triples formed from all Ha, a E S, is a set of base blocks for a (u, 3, 1)-BIBD. Now, form a u x u array R. Let 7-= { H ; :s E S } . Label the rows of R with the sets T + g , g E G F ( u ) where T + g = { H : + g } and H:+g={a+g:VaEH:}. Label the columns of R with the elements of GF (u). If a is in the adder and (b, c) is the corresponding pair, then ( b + g. c + g) is placed in cell (g, -a + g). Now. we subscript all elements in all cells of the array with 1's. Let E = {x', : 0 s t u - 1).
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101
We now form a Kirkman triple system as follows: if pair a l , bl occurs in column x’ form the triple { x i , a,, b l } . Because R is a Room square (the strong starter of Mullin-Nemeth guarantees this), all elements of the form a l b , occur once. Since every element from E occurs in each column exactly once, all pairs of the form x’a, occur exactly once. Also any pair of the form x i , xi occurs in precisely one block of { T + g :g E GF ( g ) } . The triples in row T + g along with the triples of T + g form a resolution class. Let d be a (u, 3, 1)-BIBD with variety set X and block set B. A (u, 3, 1)-BIBD D’ with variety set X’and block set B’ is said to be a subdesign of d if X ’ c X , B ‘ c B . It is easily seen that u S f ( u - 1). If equality holds, we say that D’ is a maximum subdesign. The Kirkman triple systems obtained by the Cayley construction contain maximum subdesigns. Ray Chaudhuri and Wilson [13] have also constructed Kirkman triple systems of order 2 u + 1 , u a prime power, u = 1 (mod 6), and such that each contains a maximum subdesign. If there exists a Kirkman triple system of order u containing a maximum subdesign then it can be shown [8] that this design can be used to construct a Room square of side $(u - 1). We conclude this section by showing that for all u = 3 (mod 12) and u sufficiently large, there exists a Kirkman triple system which contains a maximum subdesign. It is easily seen that a necessary condition for the existence of a Kirkman triple system of order containing a maximum subdesign is u = 3 (mod 12). The Cayley construction shows that such a design exists whenever $(u - 1) is a prime power. In order to prove that they always exist for u sufficiently large, we require the use of pairwise balanced designs. A pairwise balanced design, PBD(u; K ) , is a set B of subsets (blocks) from a u-set of elements (varieties) such that every pair of distinct varieties is contained in exactly one block of B and every block of B has cardinality belonging to the set K.
Theorem 5.1. If there exists a PBD ( u ; K ) such that for every k E K there exists a Kirkman triple system of order 2 k + 1 containing a maximum subdesign then there exists a Kirkman triple system of order 2u + 1 containing a maximum subdesign. Proof. Let Bi = {al, a,, . . . , a k } be a block of the PBD. Let the Kirkman triple system of order 2 k + 1 be written on the symbols of (2, x Bi)U {m}and denoted TB,.Without loss of generality, assume that m is not a variety of the maximum subdesign. Hence the blocks of this design which contain 0~ also contain exactly one variety in the subdesign and one not in the subdesign. Again, without loss of generality, assume that the varieties of the subdesign are (2, q),1S i S k. Since TB, is a Kirkman design, there is a resolution of the blocks into classes and each class contains a block of the form {m,(1, ai),(2, ai)}. Now, for each block of the PBD, replace the block Bi by T”,. Uniting the resolution classes which contain {m,(1, ai),(2, a i ) } ,we get a resolution of the new
{ 1 3 Y ) { 7 811) { 2 410H 8 9 12) { 3 5 I l l { 910 0) { 4 h12}{10 1 1 1) { 7 O}{ll 12 2) ( 6 K 1}{12 0 3) { 7 Y 2}{ 0 1 4) { 8 10 3}{ 1 2 5) { 911 4}{ 2 3 6 ) {to 12 5}{ 3 4 9) { I 1 0 6){ 4 S 8) ( 1 2 1 7}{ 5 6 9 ) { 0 2 X){ 6 7 10)
P
9
s
(n
?
Sfeiner systems and Room squares
103
design. If this is done for each ai,1“ - j “- u, the result is a resolution of a triple system and hence a Kirkman triple system. By the construction, it follows that if a block contains (2, ai), (2, ai), then the third variety of the block must be of the form (2, a,). Hence the constructed design contains a subdesign of order u which must be maximum. This completes the proof. By the result of Wilson [20] on PBD closure, there always exist PBD ( u ; K ) s of the type needed for Theorem 5.1 provided u is sufficiently large.
Note added in proof Since this article was written, L. Zhu of Shanghai Teacher’s College has produced a perpendicular pair of S(3,5,17)’s and C.St.J.A. Nash-Williams and A. Rosa have pointed out that in 1850 Kirkman published Cayley’s square (T.P. Kirkman, Note on an unanswered prize question, The Cambridge and Dublin Mathematical Journal V (1850) 255-262).
References [ I ] G. Beynon, Bridge Director’s Manual for Duplicate Games (George Caffin Publ., Waltham, Ma, 1943). [2] I.F. Blake and J.J. Stimer, A construction method for generalized Room squares, Aequationes Math. 14 (1976) 83-94. [3] 1.F. Blake and J.J. StiWer, A note on generalized Room squares, Discrete Math. 21 (1978) 89-93. [4] A. Cayley, On a tactical theorem relating to the triads of fifteen things, London, Edinburgh and Dublin Philos. Magazine and J. Sci. (4) 25 (1863) 59-61 (Collected Math. Papers V 95-97). [S] J.D. Horton, Variations on a theme by Moore, Proc. Louisiana Conf. Combinatorics, Graph Theory and Computing, Boca Raton (1971) 495-516. [6] E.S. Kramer, A generalized Room square GRS (4,24) of dimension 9, Discrete Math. 20 (1977) 9 1-92. [7] J.F. Lawless, Pairwise balanced designs and the construction of certain combinatorial systems, Proc. 2nd Louisiana Conference on Combinatorics, Baton Rouge (197 1) 353-366. [8] R. Mathon and S.A. Vanstone, On the existence of orthogonally resolvable Kirkman systems, Proc. 8th Manitoba Conference on Numerical Mathematics (to appear). [9] R.C. Mullin and E. Nemeth, A counterexample to a multiplicative construction of Room squares, J. Combinatorial Theory 7 (1969) 264-265. [lo] R.C. Mullin and E. Nemeth, On furnishing Room squares, J. Combinatorial Theory 7 (1969) 266-272. [ I 1 3 R.C. Mullin and W.D. Wallis, The existence of Room squares, Aequationes Math. 13 (1975) 1-7. [I21 C.D. OShaugnessy, On Room squares of order 6m +2, J. Combinatorial Theory (A) 13 (1972) 306-314. [ 131 D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman’s schoolgirl problem, Proc. Symp. Pure Math. 19 (Amer. Math. SOC.,Providence, RI, 1971) 187-203. [14] T.G. Room, A new type of magic square, Math. Gazette 39 (1955) 307. [ 151 A. Rosa, On generalized Room squares, Problkmes Combinatoires et thtorie des graphes, Orsay (1976) (Editions du CNRS, Paris, 1978) 353-358.
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[ 161 A. Rosa, Generalized Room squares and multidimensional Room designs, Proc. 2nd Caribbean Conference on Combinatorin and Computing, Cave Hill, Barbados (1977) 209-214. [17] A. Rosa, On the falsity of a conjecture on orthogonal Steiner triple systems, J. Combinatorial Theory (A) 16 (1974) 126-128. [ 181 A. Rosa and R. C. Mullin, Orthogonal Steiner systems and generalized Room squares, F’roc. 6th Manitoba Conference on Numerical Math. (1976) 315-323. [19] J.J. Stimer and I.F. Blake, An infinite class of generalized Room squares, Discrete Math. 12 (1975) 159-163. [20] R.M. Wilson, An existence theory for painwise balanced designs 11, The structure of PBD closed sets and the existence conjectures, J. Comhinatorial Theory 13 (1972) 246-273.
Annals of Discrete Mathematics 7 (1980) 105-114 @ North-Holland Publishing Company
A SURVEY OF DERIVED TRIPLE SYSTEMS K.T. PHELPS Georgia Institute of Technology, School of Mathematics, Atlanta, G A 30332, USA
1. Introduction A major problem involving t-designs is when can t-design be extended to a Except for certain special classes of t-designs, such as Steiner systems, almost nothing is known. Even for Steiner systems little has been done. However for one class of Steiner systems, namely Steiner triple systems, some progress has recently been made on this problem. It is the purpose of this article then to bring together these results, unifying this body of knowledge and generalizing wherever it is appropriate. Briefly, by a t-design S,(t, k, u ) we mean a pair (P, B)where P is a u-set and B is a collection of k-element subsets of P, usually called blocks such that every t-element subset of P occurs in exactly A blocks of B. When A = 1 the r-design is referred to as a Steiner system S(t, k, u ) . A Steiner triple system, or briefly a triple system, is an S(2, 3, u ) . Similarly a Steiner quadruple system, herein referred to as a quadruple system, is an S(3,4, n). Having established the necessary terminology we can now state more precisely the problem under consideration: When can a triple system S(2, 3, u ) be extended to a quadruple system S(3,4, u + l ) ? To clarify what is meant by extending a triple system (and in general a t-design) consider the following: given a quadruple system (P, B),if one chooses any point O E P and deletes that point from the set P and from all blocks which contain it then the resulting system (Po,B(O)),where (t
+ 1)-design?
Po=P\{O}
and B(O)={b’=b\{O}lbEB andOEb},
will be a triple system. Such a triple system is said to be derived from the quadruple system and is called, naturally, a derived triple system (or briefly a DTS). Obviously a triple system can be extended to a quadruple system if and only if it is a derived triple system. The necessary and sufficient conditions for a quadruple system S(3,4, n) to exist is that n = 2 or 4 mod 6 (Hanani [lOJ). Since the spectrum for triple systems S(2,3, u ) is that u = 1 or 3 mod 6, we conclude that there exists a DTS of every possible order. However it is unknown whether or not every triple system is a DTS. Before proceeding with our survey we remark that results on DTS that come from results in finite geometries or from finite field constructions are well 105
K.T. Pholps
106
established and in any event are of minor consequence and thus will not be considered here (see Dernbowski [7], Lindner and Rosa [14] or Luneburg rl5n. Also the interested reader should consult Cameron [ 3 ] , Dembowski [7], Witt [ 2 4 ] , or Dehon [6] for results on extensions of other t-designs.
2. Combinetorial cortstrnctions The differences between the purely combinatorial constructions presented in this section and the algebraically motivated constructions presented in the next section may not be significant in actual fact (both are quite combinatorial) but they do represent approaches which have distinctly different orientations. Associated with almost every theorem on DTS is a particular construction of a quadruple system. The construction that is needed for the theorems in this section is quite well known and can be found in Lindner and Rosa [14]. Briefly the construction goes as follows: Construction A. Let (P, B ) and (P‘, B‘) be any two quadruple systems of order n where P n P ’ = O . Let F = { F , , F 2 , .. . , Fm-,}and G = { G , ,G2,. . . , Gn-,} be any two 1-factorizations of the complete graph K , on P and P’ respectively. Let a be any permutation on the set {1,2,. . . , n - I}. Then one constructs a quadruple system ( P *, B*)with P* = P U P ’ and with, (I) BUB’CB*. ( 2 ) For any pairs x. y E P and z, w E P’, {x, y, z , w } E B* if and only if [x, y ] F,~ and [ z , W ] E G, and a ( i ) = j . Following Lindner and Rosa [14] we denote this quadruple system by ( P U P ’ )
[B.B’,F, G,a ] . As a result of t h e previous construction, we see that given an S ( 3 , 4 , n ) one can construct an S ( 3 . 4 . 2 n ) that contains the given quadruple system as a subsystem. C)with Q c P and (Briefly, a subsystem of an S(f. k, u ) . (P, B),is an S ( t , k, n ) (0, C C B.)It should also be clear from this construction that every triple system S ( 2 , 3 , 2 n - 1 ) derived from the resulting quadruple system will have a subsystem S ( 2 . 3 . n - I ) which is derived from the given S(3,4, n). The result presented next says that the converse is also true, namely:
Theorem 2.1. [ 191. A rriple sysfern S ( 2 , 3 , 2 v + 1) with a derived subsystem S ( 2 . 3 . u ) is irself derived.
Proof. Let (S, I ) be a triple system of order 2 u + 1 with derived subsystem (V, k ) of order Then every triple in f either lies completely in V or contains exactly one element of V. This induces a 1-factorization F = { F , , F 2 , . . . , F,} of S\ V as follows: [ x . y ] ~ Fiff, i E V a n d { i , z , y } E f . C h o o s e p $ S a n d l e t ( P . R ) , P = V U { p } , ti.
A suruey of deriued triple systems
107
be a quadruple system such that (V, k ) = (P,,, B(p)). Let (S\ V, B‘) be any quadruple system and G ={GI, G,, . . . ,Gv} be any 1-factorization of P where G, denotes the 1-factor that contains [i, p], for each i E V. Take a to be the identity map on {1, 2, . . . ,v} and form the quadruple system given by construction A. Then (S, t ) is a derived triple system of this quadruple system. Note that the choice of G and (S\ V, B’)in the above proof was arbitrary and thus there are potentially a large number of different S (3 ,4 ,2 v +2) that have (S, t ) as a derived triple system. For more on the above construction and its consequences see Lindner and Rosa [13,14]. The next result on derived triple system is an offshoot of the previous theorem. Before presenting this result we must again consider some techniques for constructing nonisomorphic S(3,4, n). The known methods for constructing nonisomorphic quadruple system of order n seem to fall into three general categories: use distinctly different constructions, use the same basic construction but vary the choice of components (e.g. as suggested above) or finally one can construct one S(3,4, n) and transform it into other nonisomorphic quadruple systems of order n. This latter approach has been used effectively by P. Gibbons [8] and Corneil, Mathon and Gibbons [ 5 ] . This general approach has also been used by the author [20] in the study of derived triple systems. It is for this reason that we present the following transformation of an SQS. Let (P, B) be a quadruple system with the following set of 8 blocks:
{a, b, c, 4 {a, b, x, Y} (a, c, x, 2) {a, d, Y, 2) {x, Y, 2, w } {b, c, x, w } {b, d, Y, w } {c, 4
w}. One can delete the above blocks from B replacing them with: 2,
{a, b, c, XI{a, b, d, Y} {a, c, d, z } {b,c, d, w l
Y, w,b } {x, 2, c} {Y, 2, w,4 {x, y, 2, a}. The resulting collection of blocks, B*, is again a quadruple system. One can {x,
w 7
continue to apply this transformation as long as the appropriate 8-block configurations can be found. In many cases the transformed quadruple systems will be nonisomorphic. In construction A, it should be clear that with some care in choosing 1factorizations F, G and the permutation ci one can make sure that the appropriate 8-block configurations will occur in the resulting quadruple system ( P UP’) [B, B’,F, G, a] and hence one can apply the above transformation. In fact one can choose the 1-factorizations so that the transformed quadruple system (of order 2n) has no subsystems of order n (i.e. Lindner and Rosa [13n and hence has derived triple systems that have only partial subsystems of order n. Note that (V, k ) is a partial subsystem of a Steiner system (S, t) if V c S and k = {b E t 1 b c_ V}. The next theorem is concerned with some sufficient conditions for such triple systems to be derived.
K.T. Phefps
108
Tbeorem 2.2 [20]. If (S, r) is a triple system of order 2 u + 1 containing partial subsystems (V; k ) (S\ V, r ) or order u and u + 1 respectively ( u = 1 or 3 mod 6) and if (V, k ) can be embedded in a DTS and Irl= 1, then the triple system ( S , t ) is derived. Proof. Since Irl= 1 we have J k )= % U ( U - 1)- 1 (Phelps [20, lemma 2.11). Then (V, k ) can be (uniquely) completed to a triple system (V, k U{b, c, d } ) which by assumption is a derived triple system of a quadruple system (V*, k * ) . Let us assume V * = V U { a } and that ( c , k * ( a ) ) = ( V , k U { b , c , d } )As . inTheorem 1.1 the partial subsystem (V, k ) induces a (partial) 1-factorization F = {F,, F2, . . . , F,,} of S \ V. Since Irl= 1 let r = {x, y, 2). Thus [x, y], [x, 23 and [y, z] will not be contained in any 1-factor of F but every other pair of S \ V will occur. One can assume that {b, c, x}. {b, d, y} and {c, d, z } belong to f and thus one can complete the 1-factorization F, by adding [x, y], [ x , z ] and [y, z ] to 1-factors Fb, F, and Ed respectively. (Remember fi = { [ p , q ] 1 [i, p, q ] } t,~i E V, p, q E S \ V}). Let ( S \ V, r*) be an arbitrary quadruple system with {x, y, z, w } E r * . Next let 4 : S\ V+ V U { a ) be defined as follows: (1) 4 ( x ) = d , 4 ( y ) = c, 442) = b, 4 ( w ) = a. (2) For P E S\ V, p$!{x, y, z, w } there exists a block containing [p, w ] say { i , p. W } E t. If i equals b, c, or d, then from (1) above for one choice of u E{X, y, z}, & u ) = i. Find the block containing the pair [u, w ] say G, u, w } . Set 4(p) = j . If on the other hand i does not equal b, c, or d then set 4 ( p ) = i. Clearly C#J is a one to one mapping of S \ V onto VU{a}. Applying the mapping 4 to the previously constructed 1-factorization F gives an isomorphic 1-factorization G = {GI, G,, . . . , G,} where G, is the 1-factor containing [a, i]. Then letting a be the identity, we apply construction A giving a quadruple system ( V * U S \ V) [ k * . r*, F, G. 13 which will contain the 8 block configuration: {a, b, c, d )
{a, b. x , y}
{a, c, x.
21 {a, d, Y, 2)
{x, y, z, w ) {b, c, x, w } {b, d, y, w > {c, d, z, w ) .
Next one applies the previously defined transformation to these blocks. The result is a quadruple system in which (S, 1 ) is a derived triple system. The techniques used in the above theorem can be applied to triple systems with smaller partial subsystems if enough sufficient conditions can be met (see Phelps [20]). The general idea is to reconstruct a quadruple system that has the appropriate 8-block configurations and apply a series of transformations so that the resulting quadruple system will have the desired triple system as a derived triple system.
3. Algebraic Constructions It is well known that the existence of a Steiner system S(2,3,u ) (or an S(3,4, n ) ) is equivalent to the existence of a suitable abstract algebra. In each
A suruey of derived triple systems
109
case these algebras form a variety and hence these Steiner systems are closed under the taking of direct product. It is the representations of these Steiner systems as abstract algebras that underlies the theorems in this section. However the constructions and theorems will be given a combinatorial presentation. Before proceeding we introduce some preliminary definitions and constructions. A 3-quasigroup is a pair (S, ( , ,)) where S is a set and ( ,,) a ternary operation on S such that, in the equation (x, y, z) = w if any 3 of x, y, z or w are given, the fourth is uniquely determined. The multiplication table for a 3-quasigroup is simply a latin cube. A quasigroup (S, 0 ) is similarly defined; being a binary operation its multiplication table is merely a latin square. As was mentioned above, there is a quasigroup (P, 0 ) associated with each Steiner triple system (P, B), where x o x = x for all x E P and for each pair {x, y}cP, xoy = z if and only if {x, y, z>E B. Let (Q, B,) and (P, B,) be Steiner triple systems where B , = {bl, b,, . . . , b,}. Let (Q,ol). (Q, 02), . . . ,(Q, 0 , ) be quasigroups. One can construct a triple system (Q x P, B) as follows: (1) for each block {x, y, Z}E B, and each i E P, {(x, i) (y, i) ( 2 , i)}E B ; (2) for each {i, j , k} = b, E B2 and all, not necessary distinct, elements x, y E Q we have {(x, i), (y, j ) , (x y, k)} E B where i <j < k. The triple system (Q x P, B) thus formed is called the generalized direct product of triple systems (Q, B,) and (P, B,). If in the above construction the quasigroup (Q, oi) are all identical to the Steiner quasigroup associated with (Q, B,) then (Q x P, B) is just the direct product of these triple systems. Hanani [101 showed that if (P, B,) was a derived triple system and lQl= 3, that the direct product (Q x P, B) was also derived. Subsequently B. Rokowska [22] and independently 1.5. Aliev [l], generalized this result proving that the direct product of derived triple systems is derived. In conjunction with another problem the author gave yet another special case of this result [21]. These constructions all differ to some degree. In fact B. Rokowska [22] has shown that at least in some cases her construction gives quadruple systems that are not isomorphic to those constructed by Hanani's method. What we give here is a constructive proof that is different in some respect from any of the above methods. 0,
Theorem 3.1. The direct product of derived triple systems is a derived triple system. Proof. Let (Q, B,)(P,B,) be DTS of orders n and m respectively, and let P* = PU{w}, Q*=QU{m}, wf!PUQ and let (Q*,BT) and (P * , B ;) be quadruple systems with (Q, B,) = (QZ, BT(m)) and (P,B,)= (P?, B2(w)). Let S* = (QxP)U{w} and let (QxP,B) be the direct product of (Q, B,) and (P,B,). Finally B* is the following collection of blocks: (1) {w, a, b, C}EB" for each {a, b, C}E B. This gives :nm(nm - 1) blocks (2) for all {x, y, z } E B , and {i,j, k } E B , , {(x, i ) (y, i) (2, j ) (z, k ) } E B* for all possible orderings of the triples {x, y, z} and {i, j , k}. This gives 9 ( i n ( n- 1)) (im(m - 1)) distinct blocks.
K.T.Phelps
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(3) {(x, i) (y, i) (x, j) ( y , j ) } ~ B *for all pairs {x, y } c Q and {i, j } = P, giving a total of (;)(:) blocks. (4) For each {x. y, z, w } E B : ,{x, y. z, w } c 0, {(x, i) (y, i) (z, i) ( w , i ) } E B * for all i E P giving a total of m(n(n - 1) ( n - 3)/24) blocks. (5) For each {x, y, z, W } E BY, {x, y, 2 , w } c Q with x C y C z C w and for each triple {a,, a , , a,}€ B2 where a,< a, < a 2 the following blocks are in B*: {(x. ai)(y, q)(z, 4 +i)(w,q+2)} {(x, ~ ) ( Y V q)(~ ai+z)(w, , 4+i)} {(x, q + , ) ( y ,
q+2)(2,
4)(w 4 ) ) {(x,
{x, aiMy. a , + , ) ( z ,4 ) ( w , &+,)I
{(x,
4+2)(Y,
4+1Nz, 4 ) ( w , a,)}
~ + J (qY ) k, ~ + J 4w) )
{(x, ai)(y,a , + , k4 + l ) ( w , 4)) {(x, ai+l)(y, ai>(z,q ) ( w , q+l)}
for i = 0. 1 , 2 with the subscripts reduced mod 3 as necessary. This gives 24(m(m - 1)/6)(n(n - l ) ( n - 3)/23) blocks. (6) For blocks bi EBT where bi = P, let (Q, ( , J i ) , i = 1 , 2 , . . . , m ( m - 1 ) x ( m - 3)/24, be any collections of 3-quasigroups. Then if bi ={a, b, c, d} with u < b < c < d, {(x, a)(y, b ) ( z , c)((x, y, z ) ~ d)}E , B* for x, y, z E Q where again x, y, z are not necessarily distinct. This gives n’(m(m - l)(m - 3)/24) blocks. It is easy to see that this gives the correct number of blocks and every triple of S* is contained in at least one block of B*.Hence ( S * , B*)will be a quadruple system. Applying a technique of C.C. Lindner’s [ 111 we have as an immediate corollary:
Corollary 3.2. The direct product of derived triple systems of orders
11
and m
respectively is the derived triple sysrem of at least
[ ( n - l ) ! n ! ( n - l ) ! ( n - 2 ) ! . . .2!l!]’/(nm+l)! nonisomorphic quadruple systems where r = m ( m - l ) ( m -3)/24.
Proof. There are at least (n - l)!(n!(n - l ) ! . . . 2 !l ! ) distinct 3-quasigroups of order n . As in ( 5 ) above each 3- quasigroup can be assigned independently to any of the r blocks of BT. This gives at least [ ( n - l ) ! n ! ( n - l ) ! * . *2!]’ distinct quadruple systems. Since there can be at most (nm + l)! distinct quadruple systems in any isomorphism class, the result follows immediately. ’This bound can surely be improved upon; however it suffices since this lower bound does tend towards infinity as n or m increases. In particular we have:
Corollary 3.3. The number of nonisomorphic quadruple systems of order n, n = 4 or 10 mod 18, tends toward infinity as n does.
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111
Proof. n = 4 or 10 mod 18 implies that n = 3 m + 1, m = 1 or 3 mod 6. There exists derived triple system of order m (Hanani [lo]) and therefore one of order 3m + 1 (Theorem 3.1). Since there are exactly 24 distinct 3-quasigroups of order 3, there are at least, (24)rncm- 1)(m-3)/24 /(3m + l)!
nonisomorphic quadruple systems of order n that contain this one derived triple system. It is easy to see that this lower bound goes to infinity as n does. There is no general lower bound on the number of nonisomorphic quadruple system of a given order. In particular it has not been shown that there are at least 2 nonisomorphic quadruple systems of every order greater than 10. The above corollary adds t o what is known. For more on this problem the reader is referred to Lindner and Rosa [14]. The other important algebraic construction that has been discussed in connection with the structure of derived triple systems is the singular direct product. A. Sade first introduced the singular direct product of quasigroups in 1960 [23]. Subsequently C.C. Lindner has generalized this construction and applied it to the problem of derived triple systems [12]. The definition of the singular direct product which is presented here is based on the latter’s work. Let (V, €3) be any Steiner triple system where B = {b,, b2, . . . , b,} is the collection of triples. Let (0, C ) be triple system containing the subsystem (P, C ’ ) .If p = Q \P, then for each block bi E B let (p,o i ) be a quasigroup and let S =PU(Px V). Let T be the following collection of blocks: ( 1 ) {p, q, r } E T if and only if {p, q, r} E C‘ (2) {p, (4, u ) , (r, u ) } E T if and only if p E P q, r E p, u E V and {p, q, r } E C. (3) {(p, w),(4, w),(r, W ) } E T if and only if p, q, r e p , w E V, and {p, q, r } E C. (4) {(p, u),(q, u ) , (poiq, W)}E T if and only if p , q ~ p ,bi = { u , u, W } E B and u
K.T.Phelps
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Tbeorem 3.4. If (P, B') and (V, D ) are deriued triple systems and (P, B') is an order u subsystem of a triple system (Q,C ) of order 2u + 1, then the generalized singular direct product ( S , T ) ,S = P U x V, will also be a derived triple system. Proof. Let ( P , B ) and (V, D ) be triple systems derived from quadruple systems ( P * , B*) and tV*, D*) respectively where as before P* = P U{w} and V* = V U{m}. Let (Q, C) be a triple system of order 2 [PI+ 1 which contains (P, B) as subsystem. We know by Theorem 2.1 that (Q, C) is derived. For each u E V let 0, = x { u } U P (where p = Q\P) and let C:) be a quadruple system containing ( P * , B*) as a subsystem in which (Q", C,) is a derived triple system isomorphic to (Q, C). Let S* ={a}U P U p x V and let ( S , T ) be the generalized singular direct product as described above. If (p,0 , ) is the quasigroup associated with block b, E D in that construction then we associate with each block {m} U b, E D* a 3-quasigroup (p,(,,),) where for some (fixed) element 6,~ E Fwe , have (a,x , y ) = x 0, y for all x , y E p. Such a 3-quasigroup can always be constructed; simply let (p, *) be a loop with identity element 6 and define ( x , y, z ) ~= y , z), = y z as desired (see also C.C. ( x * y ) 0, z . Since * y = y we will have (e, Lindner [12]). To the remaining blocks of D* we associate arbitrary 3-quasigroups. Now consider any one to one correspondence between the sets P* and p that maps t o 33. For each x E let x* denote the corresponding element in P*. Now we can construct T*: (1) U,,v T*. (2) { ( a ,x ) ( a ,y)(b,x)(b, y ) } T* ~ for all pairs { a , b } c and { x , y } c V. (3) For each block { x , y , z. w } , E D* with associated 3-quasigroup (F,( , ,),) where x < y < z < w and @ < u for all U E V we have { ( a ,x ) , ( b ,y),(c, z), ((a,b, c),, W ) } E T* if x f w , and {a*,(b, y ) , (c, z ) ( ( a ,b, c),, W ) } E T* if x = a for all possible triples of elements a, b. c E p, ( a , b, c not necessarily distinct). Then ( S * , T*) is a quadruple system and it will have the singular direct product ( S , T ) as a derived triple system. Note that ( S * , T*) is actually isomorphic to a suitable generalized direct product of quadruple systems (P*, B*) and (V*, D*). In closing this section we remark that necessary and sufficient conditions for the generalized direct product and the generalized singular direct product of derived triple systems to be derived still remain to be determined.
(a:,
0,
cc
4. Derived triple systems of small order
Since there exists, up to isomorphism, only one triple system of order n, n = 3 , 7, and 9 it is of course derived. For n = 13 there are two nonisomorphic triples system and N.S. Mendelsohn and S.H.Y. Hung [17] demonstrated that both are derived. The first order of real interest is 15; the smallest rigid triple system has order 15; the smallest order for which there exist 2 or more nonisomorphic Kirkman triple systems or 2 or more nonisomorphic cyclic triple
A suruey of derived triple systems
113
systems is 15; etc. For this order however it remains to be determined whether all nonisomorphic triple systems are derived. Clearly it would be of great interest to settle this question especially since for n > 15 the problem becomes computationally intractable. Cole, White and Cummings [4]first determined that there are exactly 80 nonismorphic triple systems of order 15. This was later reconfirmed by Hall and Swift [9]. A (corrected) listing of all 80 triple systems can be found in Bussemaker and Seidel [2]. The author [18] has determined that 38 of these are derived: #1-34, 61-64 (Bussemaker and Seidel [2]). P. Gibbons [8] has added #59, 70, and 76 to this list. Finally the author (unpublished) has determined that #35 and #53 are also derived. As has been amply demonstrated, a brute force computer attack on this problem is not feasible. Perhaps the most promising approach is via transformations of a given quadruple system of order 16. This idea is discussed by P. Gibbons [8] in relation to another problem.
References I.S.0. Aliev, SimmetriEeskjje algebry i sistemy Stejnera Dokl. Akad. Nauk SSR, 174 (1967) 5 11-5 13; English translation: Symmetric algebras and Steiner systems, Soviet Math. Dokl. 8 (1967) 651-653. [2] F.C. Bussemaker and J.J. Seidel, Symmetric Hadamard matrices of order 36, T.H.-Report 70WSK-02, Dept. of Mathematics Tech. University Eindhoven, Netherlands (1970). [3] P.T. Cameron, Extensions of designs: Variations on a theme, in: P.T. Cameron, ed., Combinatorial Surveys: Proceeding of the Sixth British Combinatorial Conference, (Academic Press, London, 1977) 23-43. [41 F.N. Cole, A.S. White and L.D. Cummings, Complete classification of triad systems on 15 elements, Mem. Nat. Acad. Sci., XIV (2) (1925). [ 5 ] D. Corneil, R. Mathon and P. Gibbons, Computing techniques for the construction and analysis of block designs, Utilitas Math. 11 (1977) 161-192. [6] M. Dehon, Un ThCorkme d’extension de l-designs, J. Combinatorial Theory (A) 21 (1976) 93-99. [7] P. Dembowski, Finite Geometries (Springer, Berlin, 1968). 181 P. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. Thesis, University of Toronto 1976 (Department of Computer Science, University of Toronto, Technical Report No. 92 May 1976). [9] M. Hall Jr. and J.D. Swift, Determination of Steiner triple systems of order 15, Math. Tables Aids Compat. 13 (1959) 146-152. [lo] H. Hanani, On quadruple systems, Canad. J. Math. 12 (1960) 145-157. [l 13 C.C. Lindner, On the construction of nonisomorphic Steiner quadruple systems, Colloq. Math. 29 (1974) 303-306. [12] C.C. Lindner, On the structure of Steiner triple systems derived Steiner quadruple systems, Colloq. Math. 34 (1975) 137-142. [13] C.C. Lindner and A. Rosa, There are at least 31,021 nonisomorphic Steiner quadruple systems of order 16. Utilitas Math. 10 (1976) 61-64. 1143 C.C. Lindner and A. Rosa, Steiner quadruple systems - a survey, Discrete Math. 22 (1978) 147-181. [ 151 H. Luneburg, Fahnenhomogene Quadrupelsysteme, Math. Z. 89 (1965) 82-90. [I61 E. Mendelsohn, The smallset non-derived Steiner triple system is simple as a loop, Alg. Universalis 8 (1978) 256-259. [l]
I14
K.T. Phelps
[17] N.S. Mendelsohn and S.H.Y. Hung, O n the Steiner systems S(3.4, 14) and S(4.5, 15). Utilitas Math. 1 (1972) 5-95. [IS] K.T. Phelps. Derived triple systems of order 15, M.Sc. Thesis, Auburn University, AL. 1975, S6 pp. [ 191 K.T. Phelps, Some sufficient conditions for a Steiner triple system to be a derived triple system, J. Combinatorial Theory (A) (1976) 393-397. [20] K.T. Phelps, Some derived Steiner triple systems, Discrete Math. 1 6 (1976) 343-352. [21] K.T. Phelps. Rotational Quadruple systems, A n Combinatoria, 4 (1977) 177-185. [ 2 2 ] B. Rokowska. Some new constructions of 4-tuple systems, Colloq. Math. 17 (1967) 111-121. [23] A. Sade, Produit direct singulier de quasigroups, orthogonaw et anti-atkliens, Annales de la Societe Scientifique de Bruxelles, Ser I, 74 (1960) 91-99. [24] E. Witt. Uber Steincrsche Systeme, Abh. Math. Sem. Univ. Hamburg, 12 (1938) 265-275.
Annals of Discrete Mathematics 7 (1980) 115-128 @ North-Holland Publishing Company
INTERSECTION PROPERTIES OF STEINER SYSTEMS Alexander ROSA Department of Mathematical Sciences, McMasier Unioersiry, Hamilton, Ontario, Canada, L8S 4KI
1. Introduction Given two Steiner systems of the same type S(r, k, u ) on the same u-set, how many blocks in common can they have? Can one find two such systems with no blocks in common at all? And if yes, what is the largest number of such systems with (pairwise) no common blocks that one can find? It seems that questions like these have intrigued researchers ever since the infant stages of the subject of Steiner systems. Cayley, for instance, has established in 1850 [9] that there exist two but no more disjoint Steiner triple systems on a given 7-element set, and Kirkman found in the same year that the largest number of disjoint Steiner triple systems of order 9 is seven [24]. But virtually all results of substance in this area that we summarily label “intersection properties of Steiner systems” have been obtained in the last decade. In this paper we attempt to survey the present state of affairs in this area. We also deal briefly with related questions, such as disjoint triple systems with h > 1, perpendicular Steiner systems (a subject dealt with in more detail in a paper by R.C. Mullin and S.A. Vanstone elsewhere in this volume), and applications to the existence of designs with larger A. We do not deal with questions concerning finite embeddings of (sets of) partial Steiner systems with given intersection properties; this is the subject of a paper by C.C. Lindner elsewhere in this volume.
2. Disjoint Steiner systems Two Steiner systems of type S ( t , k, u ) , say, (V, Bl), (V, B2) are disjoint if B , nB2= @, i.e. if they have no blocks in common. The symbol d(r, k, u ) will denote the maximum number of pairwise disjoint S ( t , k, u)’s. Since each S(t, k, u ) has (:)/(:) blocks, and there are altogether (ti) k-subsets of a u-set, we have d ( t , k, u)S(EI:). If the equality sign holds, the corresponding d(t, k, u ) Steiner systems are said to form a large set of disjoint S(r, k, u)’s. 2.1. Steiner triple systems The earliest results on disjoint S(2,3, u)’s are due to Cayley [9] who has shown that d(2,3,7) = 2, and Kirkman [24] who proved d(2,3,9) = 7. This result was 115
A. Rosa
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reproved by Sylvester [58,59], Walecki [40], Bays [4] and Emch [18]. However, Bays [4] was apparently the first to show that there exactly two nonisomorphic sets of seven disjoint S(2,3,9)’s, a fact verified anew in [29]. One of these sets is given by the 7 square arrays
124 128 125 129 123 126 127 378 943 983 743 469 357 346 956 765 476 586 785 489 598 while the other one is given by the square arrays
139 192 127 174 148 186 163 275 745 485 865 635 395 925 486 863 639 392 927 274 748 (the 12 triples of each system are the three rows, three columns and the six products in the expansion of the determinant of each array). The first paper that offered a significant progress on the issue in a long time and that served, in our opinion, as an impetus for further development, was that of Doyen [ 161. Although for u B 13 a few results on d(2,3, u ) date back to Kirkman (such as d(2,3. 1 3 ) a 3, d(2,3, 1 5 ) ~ [25]) 2 and a few other results were implicit in the literature (such as, say, that implied by the existence of cyclic S(2, 3, u)’s, namely that for u = 1 (mod 6), d(2,3, u ) >2), Doyen was first to offer nontrivial lower bounds for d(2,3, u ) . Let d*(t, k, u ) denote the maximum number of pairwise disjoint and isomorphic S(1, k, u)’s. Then
d*(2,3,6m + 3) 2
4m + 1 if rn = 0 , 2 (mod) 3), 4rn - 1 if rn = 1 (mod 3);
the proof is by direct construction [I61 which, in turn, uses as its basis a well-known construction due to Bose [8]. Similarly,
I’
d*(2,3,6rn + 1)2 2rn rn
if rn = O (mod 2), if rn = 1 (mod 2);
this is also shown by a direct construction based this time on a construction due to Skolem [%I. Doyen’s paper [ 161 contains also a recursive construction showing
d(2,3,2u + 1 ) a d(2,3, u ) + 2 for u 3 7
(3)
which has as its corollary
d(2.3,6rn + I ) 3 2rn - 1 for rn = 1 (mod 2).
(4)
The lower bound (1) was subsequently improved in [5] to
d*(2,3,6rn + 3) a 4m + 2.
(5)
In [26]. the inequality (4) was strengthened to
d ( 2 , 3 , 6 r n + l ) 2 3 r n + l for r n = l (mod2).
(6)
Intersection properties of Steiner systems
117
The above bounds can certainly be improved. Already in 1917, Bays [4] conjectured that
d(2,3, u ) 3 ( u - 1)/2 for all u > 7.
(7)
This has now been shown true for all u except when u = 1 (mod 12); the smallest open case is u = 3 7 . But more recently, it has been repeatedly conjectured [16,60] that d(2,3, u ) = u -2 for u 3 9 . Teirlinck [60] was the first to establish that the conjecture is true for infinitely many values of u. He proved the inequality
d(2,3,3u) 3 2u + d(2,3, u ) for every u 3 3
(8) by providing a simple and elegant recursive construction whose immediate corollary is that d(2,3,3”’)= 3“‘- 2 for all m
3 1 [60].
(9) Teirlinck’s result thus enables one to “triplicate” large sets of disjoint S(2,3, u)’s. Simultaneously with Teirlinck’s result, several direct constructions of large sets of disjoint Steiner triple systems were suggested by Denniston, Schreiber and Wilson. The most natural approach appears to be the following. Let V = Zv-;U(ml,m2} and let p = (0, 1 * * u - 3)(col)(mz) be permutation of V. The 3-subsets of V are partitioned into orbits under the action of ( p ) ; these orbits are all of full length u - 2. Suppose one can construct an S(2,3, u ) (V, B) containing exactly one triple from each orbit. For instance, if u = 13, and
B = ((0,1, X I ) , I07 2,317 (0,4951, {0,6,8), {0,7, mz), {0,9. 101, { 1 , 2 , ~ z ) , {1,3,8), {1,4,9), {1,5,10), {1,6,7), {2,4,10), {2,5,6), {2,7,m1), {2,8,9), (37 4,719 (39 5 , (39 6,101, {3,9,mz), {4,6,mz), {4,8,m i l , (5,77919 (5989 xzl, (6799 mi), (7789 101, (10, mi, mz)) m ~ ) y
(cf. [30]) then (V, B)has this property. Then obviously (V, P’B), i E &-2, is also an S(2,3, u), and p ’ B n P’B = $7 for if j . It follows that {( V, p ’ B )1 i = 0, 1, . . . , u - 3) is a large set of isomorphic disjoint S(2,3, u ) k Denniston [ 121, working by hand, succeeded in constructing in this way large sets of u - 2 disjoint S(2,3, u)’s for u = 13, 15, 19, 21,25,31,33,43,49,61,69. Moreover, an exhaustive computer search showed that there are exactly two nonisomorphic sets of 11 disjoint S(2,3, 13)’s that can be obtained in this way [12,30]. Schreiber [53] and Wilson [66] have independently shown how to construct an S(2,3, u ) with the above property provided all prime divisors p of u - 2 are such that the order of -2 modulo p is congruent t o 2 modulo 4; this condition is satisfied, for example, if p =7 (mod 8). The first few orders for which this method works are u = 9, 25,33,49, 51, 73,75, 81,91, 105, 129, 153, 163, 169, 193,201,
and for all these values of u, we have d*(2,3, u ) = u - 2. Another recursive construction - a purely combinatorial one, using properties
A. Rosa
1I X
of special 1-factorizations of K,, -appears
in [52] where it is shown that
d ( 2 , 3 , 2 u + 1 ) a u + 1+ d(2,3, u ) for u 2 7 .
(10)
This enables one to construct large sets of disjoint S ( 2 , 3 , 2 u+ 1)’s whenever there is a large set of disjoint S(2,3, u)’s. A further result on d(2,3, u ) is due to Teirlinck [62]: If u is the product of primes p for which the order of -2 ( m o d p ) is congruent to 2 (mod4) and if d(2,3, w ) = w - 2, then
d(2,3, u ( w - 2) + 2) = U( w - 2).
(1 1)
Since Teirlinck’s construction contained in [62] is not as readily available as the others discussed here, we reproduce it here in full. For u as above, the graph whose vertex-set is Z: and whose edges are the 2-subsets {g, -2g}, g E Z:, can be decomposed into two 1-factors, say F, and F2. Let {(W, B,) 1 i = 1 , 2 , . . . , w -2) be a large set of disjoint $(2,3. w)’s with W = Z,-,U{~,,~,}. Define Y = { ( V , C , j ) l i = ,l . . . , u ; j = 1 , . . . . w-2) as follows: V = ( Z , X Z , - , ) U { ~ ~ , ~ ~ } , C,,= C:,” U Cl:’ u C!;) where
C,”={P,. 3c2, (i, r)I I {=,, a;,,
Bj} U k , ( i , T ) ~ ) , (i, r2)l I k = 1 . 2 ; h,rl, r 2 } 6 Bj} U{{(i, s,), (i. sz), (i, s3)} I Is,, s2,SJEB,},
C!:)= {{xk,( r + i, s), (-2r
+ i, s + j ) } I r E Zb, s E Z,
.2 ,
( r , -2r)
E F,,
k
=
1,2}
U{{r+ i, sI),( r + i. s,), (-2r+ i, (sl + s2)/2+j)JI re Zb, s l , S,E Z ,
2 r sl
# ~217
C ! : ) = { { ( r ~ . ~ 1 ) . ( ~ ~ . ~ ~ ) , (r lr, ~ r 2,, sr 3~E+Z su ~ + j ) ) r, + r2 + rJ = 3i, r’, < r; < r;, r:, r;, r: unique integers with0sr:Su-1
and r ; + u Z = r , , k = 1 , 2 , 3 ; ~ , , s , E Z ~ - ~ } .
Then Y is a large set of disjoint S(2,3, u ( w - 2) + 2)’s. Combining all the above results and constructions, we get d(2,3, u ) = II - 2 for all u = 1 , 3 (mod 6), 9 s u ~ 2 0 5except , possibly for u = 37, 85, 97, 109, 133, 141, 145, 157, 159, 181, 195. One could require additional properties from the disjoint Steiner systems involved. Let d,(t, k. u ) be the maximum number of disjoint cyclic S ( t , k, u)’s (an S(r, k, u ) is cyclic if it has an automorphism consisting of a single cycle of length u). and denote by dr(t, k, u) the maximum number of disjoint isomorphic cyclic S(t. k, u)’s. So,for example, df(2,3,7) = 2, d,(2,3,9) = 0 (since there is no cyclic S(2,3,9)). It is easily seen that dr(2,3,6m + 1)2 2 for every m > O [16, SO]. Doyen’s construction [ 161 shows df(2,3,6m + 3) 2 4m + 1 for t+ 1 (mod 3). But not much else seems to be known about d,(2,3, u ) . One could require, as it is done in [50], that the disjoint cyclic S(2,3, u)’s have the same cyclic automorphism p ; denote the maximum number of such disjoint
Intersection properties of Sreiner systems
119
cyclic S(2,3, u)’s by d,,(2,3, u ) . Then dCc(2,3,6m + 3) = 1 (dcc(2,3,9) = 0) since any cyclic S(2,3,6m +3) must contain the same “short” orbit (one of length 2m + 1) of triples. On the other hand, dcc(2,3 , 7 ) = 2, dc,(2, 3, 13) = 2, dcc(2,3, 19) = 6 but no general results on dcc(2,3, u ) are known. Let d,(2,3, u ) be the maximum number of disjoint S(2,3, u)’s with the same regular transitive automorphism group. It is shown in [ 191 that d,(2,3,3“)= 33”- 1). A problem in a slightly different direction was proposed by Doyen [16]: Given an S(2,3, u ) with u 7, say (V, B), is there always another Steiner triple system (V, B’) isomorphic to and disjoint from it? This was recently settled in the affirmative by Teirlinck [63] who proved a much more general theorem and the best possible result in this direction.
Theorem (Teirlinck [63]). If ( V1,Bl), (V2,B2)are any two S(2,3, u)’s u 2 7, and if V is any u-set, then there exist two disjoint S(2,3, u)’s (V, B;),(V, B;)such that (Vi, Bi)-(V,B;) and (V2,B2)=(V7B!d. 2.2. Kirkman triple systems Almost one-hundred and thirty years ago, Sylvester (cf. [9,57,59]) asked the following question: “Can fifteen schoolgirls walk out in five rows of three seven times a week for a quarter of thirteen weeks in such a way that any two girls are in the same row just once in each week, and any three just once in the term?” Of course, to find such an arrangement for just one week is the famous Kirkman’s problem of fifteen schoolgirls which has “excited some attention” [9,24] in the middle of the past century. (A nice account of the problem and its modifications is by Ahrens [l], a complete classification of solutions was given by Cole [lo].) A Kirkman triple system of order u (KTS ( u ) ) is a resolvable S(2,3, u). It was only recently that Ray-Chaudhuri and Wilson [49] have shown a KTS ( u ) to exist for every u = 3 (mod 6). Let d,(2,3, u ) be the maximum number of disjoint KTS (u)’s. Then Sylvester’s problem asks whether d,(2,3, 15) = 13. This problem was assumed to have no solution by Cayley [9] (cf. also [ l , 61). Denniston [13] finally gave a solution in 1974 that was found with the aid of a computer: (0, 1,9) (0727 7) {0,3,11I {0,4,6) {0,5,8) {0,10,12) (1,495)
(2949 121 (3949 8) (1,79121 {1,8, 11) (1,273) 1395, 9) (296, 11)
(5910, 111 (576,121 {6,8,10) R 9 , 10) {6,7,9) (47 7,111 (377, 10)
(738, xi) (9911, mi} {2,5, “1) (3, 12, x , ) (4, 10, E l ) (196, mi) {8,9, 12)
{3,6,4 {1,10, ~ 2 ) {4,9, ~ 2 ) 1 5 7 , a2) (11, 12, =*I {2,8, ~02) {O, mi, mz}
This is one of the 13 KTS(l5)’s. The remaining twelve KTS’s are obtained by developing the one above modulo 13.
A. Rosa
120
Clearly dK(2,3,9) = 7 as any S(2,3,9) is resolvable. Denniston[ 111 was first to show, by recursive methods, the existence of a large set of disjoint KTS's for infinitely many orders, by proving dK(2,3,3")= 3" - 2. Schreiber [ 5 5 ] has constructed a large set of disjoint KTS (33)'s, i.e. d,(2, 3, 33) = 31. Using essentially the same method as Schreiber, Denniston [ 151 has shown d,(2,3, u ) = u - 2 for u = 51, 75, 105, 129, and subsequently, by generalizing his own recursive method of [ 113. that dK(2,3,3"m) = 3"m - 2 for n 2 1 and m = 5,25,43 [14]. Although the existence of a large set of disjoint KTS (21)'s is still in doubt, it is reasonable to conjecture, in analogy with the conjecture in Section 2.1, that dK(2,3, u ) = u 2 for u = 3 (mod 6), u 2 9 . 2.3. Steiner quadruple systems One has obviously d(3,4, u ) =su - 3. Unfortunately, apart from the trivial case u = 4, not in a single instance is the equality known to hold. It is easily established that d(3,4,8)= 2 and any two pairs of disjoint S(3,4,8)'s are isomorphic. Kramer and Mesner [29] have determined d(3,4,10) = 5 , and any two sets of five disjoint S(3.4, 10)'s are isomorphic. Only partial results are known for u a 14. The first general result in this direction seems to be the inequality d ( 3 , 4 , 2 u ) s 1+ d(3,4, u )
obtained in [38] (cf. also [39]). This was subsequently improved by Lindner [34] to d(3.4,2u) 2 u.
His construction is simple and easy to describe. Let (V, B) be an S(3,4, u ) with u 3 8, V = { 1,2, . . . , u } ; and let L be a latin square of order u with no subsquare of order 2 (such squares are known to exist for all orders u except u = 2,4). Let PI be a permutation of V defined by xp, = y if and only if the cell (x, i) of L is occupied by y. Put S = V X{ 1,2), and for i = 1,2, . . . , u, define a collection B,of quadruples as follows: B,= B:U B','where
B: = {{(x, I), (Y, I), (2,I), ( W S I , 2%{(x, 3 , (y, 2), (2,2), ( w p ; ' , 1)),
3,( w , 1)1,{(x, 21, (Y. 2), ( z p ; ' , l), (w,2% ( ( 411, (YPI, 3,( 2 , I), (w, I)}, {(x,2)- o$;', 11, (2,2), ( w , 211, ( ( 4I), (Y, I), ( Z B ,
{(XPl,
21, (Y, 1 ) s
(2,11, (w, 1)),
{(xp;'. 11, (Y, 3,(2,a,(w, 2))I {x, y, 2, W I G B), B:'= U(X9 I), (Y, l)(XS,,a,(YSI, 2)) I {x, Y) c V).
I
Then {(S,B,) i = 1,2, . . . , u} is a set of u disjoint S(3,4,2u)'s. A different construction using special 3-quasigroups and due to Phelps [48] shows d(3,4,2 5") = 5" for n 3 1.
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However, for other orders u = 2 or 10 (mod 12) it still remains to be shown that there exists at least a pair of disjoint S(3,4, u)’s! In order to construct large sets of disjoint S(3,4, u)’s, one could try an obvious extension of the method that worked well for Steiner triple systems and was described in Section 2.1. One would want to take V = Zv-3U{=1,=2,=3} as the set of elements and have p = ( ~ ~ ) ( o c ~ ) ( = ~ )1( O* * * u - 4) as an automorphism of the large set of u - 3 disjoint S(3,4, u ) k It was shown in [30] that one cannot find in this way a set of 11 disjoint S(3,4,14)’s. If one lets Cv3above to be replaced by an additive Abelian group G of order u -3 but adds the requirement that all 4-subsets {x, y, z , w} of G with x + y + z + w = 0 are contained in the initial S(3,4, u ) of a large set then it can be shown [7] that such a large set does not exist for any order u 2 8. 2.4. Orher Steiner systems. For the five Mathieu systems S(t, k, u ) (i.e. those with the Mathieu groups M , , , M,,, M22, M23, M2*, as their automorphism groups), Assmus and Mattson [2] have given a simple proof to show d(t, k, u ) 3 2 . Subsequently, Kramer and Mesner [29] have shown that for systems with the small Mathieu groups, this is the maximum number, i.e. d(4,5, 11)= 2, d(5,6, 12) = 2. Moreover, any two pairs of disjoint S(4,5, 11)’s (S(5,6, 12)’s) are isomorphic. For the systems with the large Mathieu groups, Kramer and Magliveras [28] used computer to improve the bounds substantially; they show d(5,8,24)3 9, d(4,7,23) = 24, d(3,6,22) 2 6 0 . They also show d(2,5,21)* 197; we must have d(2,5,21)<969, and Doyen [17] conjectured that d(2,5,21) = 969. Results on disjoint Steiner systems of other types are scarce. One gets d(2,4, 2,”) 3 2,”-’ - 1 immediately by taking into account that the S(3,4, 2’”) -AG(2n, 2) can be partitioned into S(2,4, 2 2 n ) ’[3,67]. ~ Magliveras [41] has shown d(2,4, 1 3 ) 2 3 3 but it is not known whether this is best possible (maximum possible is 55). It follows from [45] that d ( 3 , 5 , 1 7 ) s 2. A general result analogous to that of Teirlinck [63] in Section 2.1 is that if (V,, B,),(V,, B2)are two S(t, k, u)’s with 2t s k < u, then there exist two disjoint S(t, k, u)’s (V, B;),(V, B:) such that (V,, B,)= (V, B;) and (V,, B2)= (V, B:) [20]. Thus d(t, k, u ) 3 2 whenever there exists an S(t, k, u ) with 2t s k < u.
3. Steiner systems with prescribed intersections 3.1. Almost disjoint Steiner triple systems. In [16], Doyen asked, among other things, the following question: Does there exist for every order u 3 3 a pair of S(2,3, u)’s having exactly one triple in common? That the answer is yes was shown by Lindner [32,33]. One can, of course, extend the problem in various directions. In [36], the existence of large sets of mutually almost disjoint (MAD) S(2,3, u)’s is considered. Here, by a large
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set of S(2.3, u)’s is meant a set {( V, Bi)I i E I } of S(2,3, u)’s such that UiE,Bi = (y), the set of all 3-subsets of V. A set of MAD S(2,3, u)’s is maximal if it is not a proper subset of some set of MAD S(2,3, u)’s. A construction that uses the existence of S(3,4, u ) for every order u = 2 or 4 (mod 6) is given in [36] to show the existence, for u 5 9, of a large set of MAD S(2,3, u)’s. For u 2 15, every large set of MAD S(2,3, u)’s is maximal. For u = 7, there are large sets of MAD S(2,3,7)’s containing m systems for m E {7,8, . . . , 15). For u = 9, there exist maximal sets of 4, 5 or 9 MAD S(2,3,9)’s,there is up to an isomorphism only one large set of MAD S(2,3,9)’s (containing, of course, 9 systems). For u = 13, a large set of MAD S(2,3, 13)’s must contain 13, 14 or 15 systems. It is not known whether 14 or 15 are possible. It was shown in [36] that for u 2 15, a large set of MAD S(2,3, u)’s must contain u - 1, u or u + 1 systems. That u - 1 is impossible was shown subsequently in [64]. Whether there exists a large set of u + 1 MAD S(2,3, u)’s is an open question. Similar questions can be asked for resolvable S(2,3, u)’s. It was shown in [23] that for all u = 3 (mod 6) there exists a pair of almost disjoint Kirkman triple systems but I am not aware of any further results in this direction.
3.2. Steiner systems with given number of blocks in common Two S ( t , k, u)’s (V, B,),(V, €3,) are said to intersect in m blocks if (B, nBl= m. Denote by J [ t , k. u] (by J*[t, k, u ] , respectively) the set of all integers m such that there exists a pair of S ( t , k, u)’s (a pair of isomorphic S ( t , k, u)‘s, respectively) intersecting in m blocks. In [37], the sets 5[2,3, u ] are completely determined. Let I , = ‘,u(u - l), and let 1, = (0, 1, . . . , 1, - 6, t, - 4, t,}, i.e. the set I, contains all nonnegative integers not exceeding t, with the exception of the four numbers 1 , - 5 , 1, - 3, f, - 2, I , - 1 . Then 5[2,3,7] = 5*[2,3,7] = (0, 1,3,7}, 5[2,3,9] = 5*[2.3,9] = {O, 1 , 2 , 3 , 4 , 6 , 12}, and for u 2 13, 5[2,3, u] = I,, [37]. On the other hand, to determine t h e sets 5*[2,3, u ] for u 2 - 13 is an open problem. Similarly, if JR[t. k. u ] denotes t h e set of all integers m such that there exists a pair of resolvable S(f. k, u)’s intersecting in rn blocks then it follows from Teirlinck’s result [63] of Section 2.1, and from Hall and Udding’s result [23] of Section 3.1, respectively, that for u = 3 (mod 6), O~5,[2,3, u] for u 29, and 1 ~ 5 , [ 2 , 3 ,u] for u 2 3 , but for u 3 15, to determine 5,[2,3, u] is an open problem. As for Steiner systems other than triple systems, one finds easily 5[3,4,8] = 5*[3.4,8] = {0,2,6, 14). It is shown in [29] that 5[3,4, 101 = 5*[3,4, 101= { 0 , 2 , 4 , 6 , 8 , 12, 14,30}, 5[4,5, 11]=5*[4,5,11]={0,6, 12, 18,30,66}, 45,6,12]=5*[5,6, 12]={0, 12,24,36,60, 132). To best of our knowledge, the set J[r, k, u] remains undetermined for any other
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(nontrivial) parameter set for which an S ( t , k, u ) is known to exist. There are, however, several partial results, such as
(0, L21= 5*[3,6,221, {O, 1 , 2 , 3 , 4 } ~ 5 * [ 4 , 7 , 2 3 1 , ( 0 , 1 , 2 , 3 , 4 , 5 , 7 , 9 } c J * [ 5 , 8 , 2 4 ] [281. A slightly different question was asked by Doyen in [ 161: what is the maximum number d(2,3, u ; m ) of S(2,3, u)’s on a given u-set such that any two of them intersect in m triples, these m triples occurring, moreover, in each of the S(2,3, u)’s? It follows from what was said in this section that d(2,3, u ; m )3 2 whenever m E 5[2,3, u] but almost nothing more seems to be known about the function d(2,3, u ; m). Of course, similar questions can be asked about other Steiner systems S ( t , k, u ) .
3.3. Steiner systems intersecting in subsystems Following [22,23], define, for a given partial triple system of order m T = (S, B), a (u, T)-n-tuple of S(2,3, u)’s to be a set {(V, Ai) I i = 1 , . . . , n} of S(2,3, u)’s such that S c V and Ai nAi = B for i f j. If T happens to be an S(2,3, m ) then one speaks of a (u, m)-n-tuple of S(2,3, u)’s. The following theorem is proved in [22]:
Theorem. If u 3 2 m + 1 and (u, m ) # ( 3 , l), then there exists a (u, m)-pair of S(2,3, u)’s. Observe that this result is obviously best possible, and that the cases m = 1 and m = 3 correspond to the case of disjoint, and almost disjoint S(2,3, u)’s, respectively, that were considered earlier. Observe also that because of the replacement property of Steiner triple systems, the above result implies that given any S(2,3, m ) (S, B), and u a 2m + 1, (u, m) # (3, l), there exists a pair of S(2,3, u)’s (V, A , ) , (V, A,) with S c V and A, f l A, = B. Concerning partial triple systems, it is shown in [23] that if T = ( S , B) is a partial triple system of order m, and if either u = 3 (mod 6) and u z=6m + 3 or u = 1 (mod 6) and u 3 12m + 7, then there exists a (u, T)-pair of S(2,3, u)’s. Unlike in the previous case, this result can certainly be improved. For more detailed discussion of some specific partial triple systems, notably so-called claws, see [23].
4. Related questions
4.1. Disjoint triple systems with A 3 1 A triple system SA(2,3, u ) is a pair (V, B) where V is a u-set and B is a set of 3-subsets of V called triples such that any 2-subset of V is contained in exactly A
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triples. It is well-known that a necessary condition for the existence of an SA(293, u ) is A = O (mod A(u)) where A(u) = 1 if u = 1 or 3 (mod 6),
A ( u ) = 2 if u=O-or 4 (mod6), (*)
A ( u ) = 3 if u = 5 (mod6),
A(u)=6
if u = 2 (mod6).
(It is also well-known that if one does not require all elements of B to be distinct as subsets then this condition ( * ) is also sufficient.) Let d , ( 2 , 3 , u ) be the maximum number of pairwise disjoint S ( 2 , 3 , u)’s. Trivially d,(2,3, u ) S ( u - - 2 ) / A . Teirlinck conjectured [61] that d , , , , ( 2 , 3 , u ) = ( u - 2)/A(u) for all u # 7. The case A(u) = 1 was discussed, of course, in Section 2.1. Schreiber [54] has proved d,(2,3, u ) = + ( u -2) for u = O or 4 (mod 12). Teirlinck [61] (see also [62]) has proved
d z ( 2 , 3 , u ) = $ ( u - 2 ) for all u = O or 4 (mod6), u > O and
d , ( 2 , 3 , u ) = b(u - 2) for all u = 2 (mod 6), and so, for u even, dA,,,(2,3 , u ) is completely determined. For A(u) = 3 there are two partial results. First, there is a construction by Kramer [27] showing d 3 ( 2 ,3 , u ) = f ( u - 2 ) whenever u is a prime power, u = 5 (mod6). Another construction is due to Teirlinck [62]; he shows that if u is the product of primes p for which the order of -2 (mod p ) is congruent to 2 (mod 4), then d 3 ( 2 ,3 , 3 u + 2 ) = u. (Here necessarily 3u + 2 = 5 (mod 6) as for u as above, u = f 1 (mod 6).) The smallest values of u = 5 (mod 6) that are not covered by either of the two constructions, and for which, consequently, the question whether d3(2, 3, u ) = f(u-2) remains open, are u = 35, 65, 77, 119, 155, 161, 185, 203.
4.2. Existence of designs S,(r, k, u ) . A r-design S,(L k, u ) is a pair (V, B) where V is a u-set and B is a set of k-subsets of V called blocks such that any r-subset of V is contained in exactly A blocks. Clearly, the existence of A disjoint S ( t , k, u)’s implies the existence of an S,(r, k, u ) , and also of an S,dt, k, u ) where A ’ = (:::)-A. Thus, for instance, the results of [28] surveyed in Section 2.4 imply that there exists an S,(5, 8,24) for l s A s 9 , and S,(4,7,23) for 1 s A s 2 4 , an S,(3,6,22) for 1 s A s 6 0 and an S, (2,5,2 1) for 1s A S 197, and also for A ’ = 969 - A whenever A is in the above range. More generally, if there exist s disjoint S,(r, k, u)’s, then there exists an SsA( r, k, u ) . If a design S, (2,3, u ) exists then 1=sA =s u - 2. Therefore if we want to show that the designs S,(2, 3 , u ) exist for all values of A in this range, it suffices to show the existence of at least s, = [ ( v - 2)/(2A(u))] disjoint S,,,,(2, 3 , u)’s. It
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follows from what was said in Section 2.1 and Section 4.1 that this has already been proved for u even, and for u = 3, 7 or 9 (mod 12) (this time including also u = 7!) but not for u = 1 (mod 12) (when h ( u ) = 1) or for u = 5 (mod 6) (when h ( u ) = 3). Observe that assuming the existence of at least s, disjoint SA,,,(2,3, u)’s in these two cases is a much weaker assumption than either the conjecture of Section 2.1 or Teirlinck’s conjecture of Section 4.1. 4.3. Perpendicular Steiner svstems One can put additional requirements on disjoint Steiner systems; the motivation for this may come from an area outside of the theory of block designs. Two Steiner systems (V, Bl), (V, B2) of type S(t, k, u ) are perpendicular if (i) they are disjoint, (ii) whenever Q, Q are two blocks of Bl intersecting in a (k - t)-subset P, and R, R‘ are such that ( Q \ P ) U R E B ~ (, Q t \ P ) U R ’ ~ B 2then R f R’. Perpendicular Steiner systems (older term: orthogonal) were introduced in [45] as a generalization of perpendicular S(2,3, u)’s that are dealt with in [21,35,42,43,44,46,47,51]. The relationship between the perpendicular S(r, k, u)’s and the existence of Room squares and their generalizations is discussed in detail in a paper by R.C. Mullin and S.A. Vanstone elsewhere in this volume. The main open problem here is that of the existence of pairs of perpendicular S(2,3, u)’s. Let dp(2,3, u ) be the maximum number of pairwise perpendicular S(2,3, u)’s. A construction in [43] shows dp(2,3, u ) =- 2 whenever u = p” = 1 (mod 6) is a prime power. The set of orders u = 1 (mod 6) for which dp(2,3, u ) 3 s is PBD-closed [65]; this was shown for s = 2 (and any u ) by Lawless [31] and for any s by Gross [21]. Different constructions for pairs of perpendicular S(2,3, u)’s where u = p a = 1 (mod6) is a prime power were given in [42]. It was shown in [35] that the existence problem for pairs of perpendicular S(2,3,u)’swith u = 1 (mod 6) can be reduced to solving the problem for the case u = p * q where p and q are primes = 2 (mod3). The smallest order u = l (mod6) for which a pair of perpendicular S(2,3, u)’s is not known, is u = 55. As for orders u = 3 (mod 6), there exists no pair of perpendicular S(2,3,9)’s (cf. [44]). It was conjectured [45,46] that there does not exist a pair of perpendicular S(2,3, u)’s for any u = 3 (mod 6). However, in [51] a pair of perpendicular S(2,3,27)’s was constructed. It follows easily that dp(2,3, u ) a 2 for u = 1 or 3 (mod 6) with finitely many exceptions, i.e. there exists a constant u,(2) such that dp(2,3, u ) b 2 for all orders u > u,(2), u = 1 or 3 (mod 6). The smallest orders u = 3 (mod6) for which it is not decided whether a pair of perpendicular S(2,3, u)’s exists, are u = 15,21,33,39,45,51,57,63,69,75. Similarly, there exists a constant u;(s) such that for all orders u = 1 (mod 6), u > u;(s), dp(2,3, u ) 2 s. A table for orders u =s175 with best known lower bounds for 4 ( 2 , 3 , u ) appears in [21]. In particular, dp(2,3,31) 2 6 ; u = 31 is the smallest order for which a set of more than two perpendicular S(2,3, u)’s is known to
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exist. Recently, Zhu [68] has provided a new direct construction for sets of perpendicular S(2.3. u ) ‘ s . In particular. h e shows d,(2,3.2” - 1 )3 3 whenever (n,6) = 1, and dJ2.3. 127)5 6. Clearly. d,(2. 3 . u ) s u-2. and we have d,(2,3,7)= 2, d,(2, 3, 9 ) = 1. It is widely believed that d,(2, 3, u ) s i ( u - 1 ) but n o nontrivial upper bounds on d,(2. 3. u ) are known. Not much is known about perpendicular Steiner systems other than S(2,3, 111’s. The following condition is necessary for the existence of a pair of perpendicular S(1, k, u)’s [45]:
For instance, when r = 3. k = 4, this forces u S 8 ; there exists indeed a pair of perpendicular S(3,4,8)’s [45]. Thus, d,(3,4, u ) =
2
if u = 8,
1
if v
2 or (mod 6), u f 8.
Similarly, d,(4, 5, u ) = 1 for all u for which an S(4,5, u ) exists [45]. On the other hand, d,(4.7,23) 2 9 (cf. [28,45]). Observe that directly from the definition of perpendicular Steiner systems follows that if k 2 2 t , any two disjoint S(r. k . u)’s are perpendicular.
References W. Ahrens. Mathematische Unterhaltungen und Spiele (B.G. Teubner. Leipzig, 19 18). E.F. Assnius Jr. and H.F. Mattson Jr.. Disjoint Steiner systems associated with the Mathieu groups. Bull. Amcr. Math. Soc. 72 (1966) 843-845. R.D. Baker. Partitioning the planes of AG2,,,(2) into 2-designs. Discrete Math. 15 (1976) 20.5-2 1 I . S. Bays. llne question de Cayley relative au probleme des triades de Steiner, Enseignement Math. 19 (1917) 57-67. G.F.M. Beenker. A.M.H. Gerards and P. Penning, A construction of disjoint Steiner triple systems, T H Report 78-WSK-01, Department of Mathematics, Technological Univ. Eindhoven. C. Berge. ThCorie des Graphes et ses Applications (Dunod, Paris. 19.58). T. Beth. On resolutions of Steiner systems. Dissertation. Erlangen 1978. K.C. Boss. On the construction of balanced incomplete block designs. Ann. Eugenics 9 (1939) 353-399. A . Cayley. On the triadic arrangements of seven and fifteen things, London, Edinhurgh and Dublic Philos Mag. and J. Sci. (3) 37 (18.50)50-53. F.N. Cole. Kirkman parades. Bull. Amer. Math. SOC.28 (1922) 43.5-437. R.H.F. Denniston. Double resolvability of some complete 3-designs. Manuscripta Math. 12 11974) 105-1 12. R.H.F. Denniston. Some packings with Steiner triple systems, Discrete Math. 9 (1974) 213-227. R.H.1:. Denniston, Sylvester’s problem of the 15 schoolgirls, Discrete Math. 9 (1974) 220-23-1. R.H.F. Denniston. Further cases of double resolvability, J. Combinatorial Theory ( A ) 26 (1Y7Y) 79X-3(13 K . H . F . Dennistun. Four doubly resolvable complete designs, Ars Combinatoria 7 ( 1979) 265272.
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[16] J. Doyen, Constructions of disjoint Steiner triple systems, Proc. Amer. Math. SOC.32 (1972) 409-4 16. [17] J. Doyen, Recent developments in the theory of Steiner systems, Teorie Cornbinatorie I, Colloq. Roma 1973, Atti dei Convegni Lincei 17, Roma (1976) 277-285. [18] A. Emch, Triple and multiple systems, their geometric configurations and groups, Trans. Amer. Math. SOC.31 (1929) 25-42. [ 191 G. Ferrero, Su un problema relativo ai sistemi di Steiner disgiunti, Rend. 1st. Mat. Univ. Trieste 7 (1975) 1-7. [20] B. Ganter, J. Pelikan and L. Teirlinck, Small sprawling systems of equicardinal sets, Ars Combinatoria 4 (1977) 133-142. [21] K.B. Gross, On the maximal number of pairwise orthogonal Steiner triple systems, J. Combinatorial Theory (A) 19 (1975) 256-263. [22] J.I. Hall and J.T. Udding, On pairs of Steiner triple systems intersecting in subsystems, Technological University Eindhoven Report 76-WSK-04, August 1976. [23] J.I. Hall and J.T. Udding, On intersection of pairs of Steiner triple systems, Indag. Math. 39 (1977) 87-100. [24] T.P. Kirkman, Note on an unanswered prize question, Cambridge and Dublin Math. J. 5 (1850) 255-262. [25] T.P. Kirkman, Theorems on combinations, Cambridge and Dublin Math. J. 8 (1853) 38-45. [26] A. Kotzig, C.C. Lindner and A. Rosa, Latin squares with no subsquares of order two and disjoint Steiner triple systems, Utilitas Math. 7 (1975) 287-294. [27] E.S. Kramer, Some triple system partitions for prime powers, Utilitas Math. 12 (1977) 113-1 16. [28] E.S. Kramer, S.S. Magliveras, Some mutually disjoint Steiner systems, J. Combinatorial Theory (A) 17 (1974) 39-43. [29] E.S. Kramer, D.M. Mesner, Intersections among Steiner systems, J. Combinatorial Theory (A) 16 (1974) 273-285. [30] E.S. Kramer, D.M. Mesner, The possible (impossible) systems of 11 disjoint S(2.3, 13)'s (S(3.4, 14)'s) with automorphism of order 11, Utilitas Math. 7 (197.5) 55-58. 13 13 J.F. Lawless, Pairwise balanced designs and the construction of certain combinatorial systems, Proc. 2nd Louisiana Conf. Combinatorics, Graph Theory and Computing, Baton Rouge (1971) 353-366. [32] C.C. Lindner, A simple construction of disjoint and almost disjoint Steiner triple systems, J. Combinatorial Theory (A) 17 (1974) 204-209. [ 3 3 ] C.C. Lindner, Construction of Steiner triple systems having exactly one triple in common, Canad. J. Math. 26 (1974) 225-232. [34] C.C. Lindner, A note on disjoint Steiner quadruple systems, Ars Combinatoria 3 (1977) 27 1-276. [35] C.C. Lindner and N.S. Mendelsohn, Construction of perpendicular Steiner quasigroups, Aequat. Math. 9 (1973) 150-156. [36] C.C. Lindner and A. Rosa, Construction of large sets of almost disjoint Steiner triple systems, Canad. J. Math. 27 (1975) 256-260. [37] C.C. Lindner and A. Rosa, Steiner triple systems having a prescribed number of triples in common, Canad. J. Math. 27 (1975) 1166-1175. Corrigendum: 30 (1978) 896. [38] C.C. Lindner and A. Rosa, Finite embedding theorems for partial Steiner quadruple systems, Bull. SOC.Math. Belg. 27 (1975) 315-323. [391 C.C. Lindner and A. Rosa, Steiner quadruple systems-a survey, Discrete Math. 22 (1978) 147-18 I . [40] E. Lucas, RCcrCations mathtmatiques, Vol. 2 (Gauthier-Villars, Paris, 1883). [41] S. S. Magliveras, Private communication. [42] N.S. Mendelsohn, Orthogonal Steiner systems, Aequat. Math. 5 (1970) 268-272. [43] R.C. Mullin and E. Nemeth, On furnishing Room squares, J. Combinatorial Theory 7 (1969) 266-272. [44] R.C. Mullin and E. Nemeth, On the nonexistence of orthogonal Steiner systems of order 9, Canad. Math. Bull. 13 (1970) 131-134. [45] R.C. Mullin and A. Rosa. Orthogonal Steiner systems and generalized Room squares, Proc. 6th
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Manitoba Conf. Numerical Math., Congressus Numerantium XVlIl (Utilitas Math., Winnipeg, 1977) 315-323. [46] C.D. OShaughnessy, A Room design of order 14. Canad. Math. Bull. 11 (1968) 191-194. [47] C.D. OShaughnessy, On Room squares of order 6 m +2, J. Combinatorial Theory (A) 13 (1972) 306314. 1481 K.T. Phelps. A construction of disjoint Steiner quadruple systems, Roc. Eighth S.-E. Conf. Combinatorics, Graph Theory and Computing, Baton Rouge 1977, Congressus Numerantium XIX (Utilitas Math., Winnipeg, 1977) 559-567. [49] D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman's schoolgirl problem, Proc. Sympos. Pure Math. 19 (Amer. Math. Soc., Providence, RI, 1971) 187-203. [50] A. Rosa, P o z n h k a o cyklick9ch Steineroech systkmoch trojic, Mat.-Fyz. cas. 16 (1966) 28.5-290. [51] A. Rosa, On the falsity of a conjecture on orthogonal Steiner triple systems, J. Combinatorial Theory (A) 16 (1974) 126-128. [S2] A. Rosa, A theorem on the maximum number of disjoint Steiner triple systems, J . Combinatorial Theory (A) 18 (1975) 305-312. [S3] S. Schreiber, Covering all triples on n marks by disjoint Steiner systems, J . Combinatorial Theory (A) 15 (1973) 347-350. [54] S. Schreiber, Some balanced complete block designs, Israel J. Math. 8 (1974) 31-37. [55] S. Schreiber, Private communication. [56] Th. Skolem. Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958) 273-280. [57] J.J. Sylvester, Note on the historical origin of the unsymmetrical six-valued function of six letters, London, Edinburgh and Dublin Philos. Mag. and J. Sci. 21 (1861) 369-377. [58] J.J. Sylvester, Remark on the tactic of nine elements, London Edinburgh and Dublin Philos. Mag. and J . Sci. (4) 22 (1861) 144-147. [59] J.J. Sylvester, Note on a nine schoolgirls problem, Messenger Math. (2) 22 (1892-93) 159-160, Correction: 192. [60] L. Teirlinck, On the maximum number of disjoint Steiner triple systems, Discrete Math. 6 (1973) 299-300. [61] L. Teirlinck, On the maximal number of disjoint triple systems, J. Geometry 6 (1975) 93-96. [62] L. Teirlinck, Combinatorial Structures, Thesis, Vrije Universiteit Brussel, Departement voor Wiskunde, 1976. [63] L. Teirlinck. On making two Steiner triple systems disjoint, J. Combinatorial Theory (A) 23 (1977) 349-350. [64] T.M. Webb, Some constructions of sets of mutually almost disjoint Steiner triple systems, M.Sc. Thesis, Auburn Univ. 1977. [65] R.M. Wilson, An existence theory for painvise balanced designs, I. J. Combinatorial Theory (A) 13 (1972) 220-245; 11: 13 (1972) 246-273. [66] R.M. Wilson, Some partitions of all triples into Steiner triple systems, Hypergraph Seminar, Ohio State Univ. 1972, Lecture Notes Math. 411 (Springer, Berlin, 1974) 267-277. [67] G.V. Zaicev, V.A. Zinoviev and N.V. Semakov, Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes, Proc. 2nd Internat. Sympos. Information Theory, Tsahkadsor, Armenia, USSR, 197 1 (Akadkmiai Kiado, Budapest, 1973) 257-263. [68] L. Zhu, A construction for orthogonal Steiner triple systems, Ars Combinatoria (to appear).
PART IV
RESOLVABILITY AND EMBEDDING PROBLEMS FOR STEINER SYSTEMS
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Annals of Discrete Mathematics 7 (1980) 131-141 @ North-Holland Publishing Company
ON IDENTIFYING PG(3,2) AND THE COMPLETE 3-DESIGN ON SEVEN POINTS J.I. HALL Deparfmenf of Mathematics, Michigan Sfare Uniuersiry, East Lansing, MI 48824, USA
Let f be a graph whose 35 vertices are labelled by the distinct 3-element subsets of a set of size 7. We join two vertices of f whenever their labels intersect in a set of size 1. Let f* be a graph whose 35 vertices are labelled by the 35 distinct lines of a projective geometry of dimension 3 over GF(2). We join two vertices of f* whenever their labels are intersecting lines. Then as graphs f and r* are isomorphic. In this note we verify the graph isomorphism mentioned above and thereby identify the projective geometry PG(3,2) with the complete 3-design on 7 points. This identification then allows us in an elementary fashion to prove various results related to both designs. For instance, by investigating aut(r) = aut(f*) we verify the sporadic isomorphism A, = GL(4,2). By examining the complete 3-design on 7-points, we are able to catalogue all parallel classes and Kirkman parallelisms of PG(3,2). Conversely, the geometry of PG(3,2) aids us in finding various resolutions of multiples of the complete 3-design on 7 points. Much of what we shall discuss is relatively well-known (indeed Section 5 reproves several results given in [ 11). Our advantage is that the results are proved here in an elementary manner.
1. PG(2,2) and PG(3,2) We shall be concerned mainly with three designs - the projective spaces PG(2,2) and PG(3,2) and the complete 3-design on 7 points Here we may think of PG(2,2) as the design whose points are the non-zero vectors of a three dimensional vector space over GF(2) and lines are the sets of three non-zero linearly dependent vectors. PG(3,2) is the design whose points are the non-zero vectors of a 4-dimensional space over GF(2), lines are the sets of three non-zero linearly dependent vectors, and hyperplanes are the sets of seven non-zero vectors of subspaces of dimension 3. We shall let X = { 1,2,3,4,5,6,7}, so that the design is all subsets of X with cardinality 3 and has parameters t - ( u , k, A ) with t = k = 3, u = 7, and A = 1. General properties and definitions concerning designs and incidence structures can be found in [3], [4]. We shall sometimes consider our designs as subsets of the point set and other times as incidence structures, in the interest of clarity.
(T).
(T)
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1.1. Hall
132
While the only properties of (f) and its automorphism group S, that we shall need are evident, certain well-known and easily verified properties of PG(2,2) and PG(3,2) will be assumed.
1.1. Aut(PG(2,2)) is the group G L ( 3 , 2 ) s A , and has order 168. GL(3,2) is doubly transitive on the points and lines of PG(2,2). No non-identity element of GL(3,2) fixes more than three points.
1.2. Aut(PG(3,2)) is the group GL(4,2) of order $(8!).
13. Let Y be a set of size 7 or 8. Further let d be a collection of subsets of Y , each containing three elements and maximal subject to having pairwise intersections of size 1. Then d is the set of lines of a design on 7 of the points of Y which is isomorphic to PG(2,2). In particular, up to isomorphism, there is a unique 2-(7,3, 1) design. Note that the points and lines of the hyperplanes of PG(3,2) form a design PG(2,2). Conversely, this characterizes PG(3,2).
1.4. A 2-( 15,3, 1) design is isomorphic to PG(3,2) if and only if each collection of three points of the design are points of a subdesign isomorphic to PG(2,2). This is a consequence of Pasch’s axiom “31, p. 24, (3)]. Alternatively, the information given is enough to reconstruct the underlying vector space easily. 2. Identifying (f)and PG(3,2)
(t)
In this section we define an incidence structure .9 on which is seen to be the structure of PG(3,2). Let G be the symmetric group S7 on { 1 , 2 , 3 , 4 , 5 , 6 , 7 } ,and let K be its unique subgroup of index 2, K = A 7 . We choose a particular design P from ):( and isomorphic to PG(2,2):
P = { ( I , 2,3), ( 2 , 4 , 7 ) ,(3,4,6), ( 1 , 4 , 5 ) ,(2,5,6), ( 3 , 5 , 7 ) ,(1,6,7)). Note that we are denoting members of (f) by triples (a, b, c). Now H = G{p)= GL(3,2) has order 168, hence P has 30 distinct images under G, that is, there are precisely 30 distinct PG(2,2) from (f). As H S K , the 30 distinct PG(2,2) fall into two K-orbits, each of length 15. Let one of these obrits be denoted +c, and call its members points. Let the other orbit be denoted Ye, and call its members hyperplanes. Call the 35 triples of lines. We define now an incidence structure 9 on /cU (T) U X If t‘E (f) and Y ejz U X, then 8 and Y are incident if and only if the triple 4 is contained in the PG(2,2) design Y. If YE/^ and Q E 2,then Y and Q are incident if and only if the
(T)
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(T).
intersection Y n Q of the two PG(2,2) contains at least one line from No other incidences occur. Observe that K acts in a natural fashion on the incidence structure preserving /z, and X and respecting incidence. The group of all permutations of /zU U X which do this forms the automorphism group of 4 and so contains K. Note that K is transitive on each of 6, and X. A permutation of 4 which respects incidence and but exchanges the sets z/, and X is called a duality of 4. It is clear that all elements of G-K induce dualities of 4. We say that two lines meet whenever they are incident to a common point or a common hyperplane.
(t), (t)
(T)
(T),
2.1. Two lines t', and 8, meet if and only if It',nt',l
= 1. If two lines meet then there is a unique point and a unique hyperplane incident with both lines.
Proof. Clearly if 4, and t', meet, then It', fl[,I=
1. Now suppose It',fl t',l = 1. By the transitivity of K we may assume 8, = (1,2,3) and 4, = (1,4,5). These two lines can be completed to a PG(2,2) in exactly two ways - as the design P given above and as Pa where a = (6,7) E G. As a is a duality, exactly one of P and Pa is a point and the other is a hyperplane. 2.2. (1) Each line is incident with exactly 3 points and exactly 3 hyperplanes. (2) Any two points are incident to exactly one common line. Any two hyperplanes are incident to exactly one common line. (3) If Y is a point and Q is a hyperplane with Y and Q incident, then there are exactly 3 lines incident with both Y and 0. (4) Each hyperplane is incident with exactly 7 points and 7 lines.
Proof. By the transitivity of K we need prove (1) only for the line t'= (1,2,3). Let S be the set of all PG(2,2) from U X which contain 4'. As t'r~P and H is transitive on the lines of P, S ={PgI g e G{el}. Thus IS(= IG{,,I/JG{,,nHI= 144/24 = 6. We have in fact that S ={PIg E ((1,2,3), (4,5))}. This gives (1). Observe that P f l Pg= t' for g = ( 1 , 2 , 3 ) and g = (1,3,2); thus no pair of points is incident to more than one line, and dually no pair of hyperplanes is incident to more than one line. Using (l),each pair of points is incident on the average to 35-3/(:5)= 1 line. This and a dual observation give (2). For (3) assume that t'= (1,2,3)E Y n Q and P is one of Y and Q. If P is a point (or hyperplane) incident with 8 then the three hyperplanes (or points) incident with 't are P(a, 6) for (a, 6) E G, {a, 6 )G { 1,2,3}. We find that P f P( l a, 6) consists precisely of the three lines of P containing c where {a, b, c} = {1,2,3}, giving (3). As Ifi = 1x1= 15, (4) is a direct consequence of (1) and (3). 2.3. Each triple of points of 4 is incident with at least one hyperplane of 4. The points and lines incident with a given hyperplane form a PG(2,2).
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Proof. If the triple of points is collinear (incident to a common line) then the first sentence follows from 2.2(1). Otherwise it is a consequence of 2.1 and 2.2(2). From 2.2(4) each hyperplane is incident with 7 points and 7 lines. 2.2(2) and 1.3 then imply that these points and lines form a PG(2,2), as desired. Now 1.4, 2.2(1), 2.2(2), and 2.3 give immediately 2.4. The incidence structure 9 is isomorphic to the incidence structure of points, lines, and hyperplanes of the projective geometry PG(3,2).
3. The graph
r
r
Recall that the graph is a graph with 35 vertices, each labelled by a member of ):( and two vertices joined whenever their labels have intersection of size 1. In are terms of our correspondence with PG(3,2), 2.1 states that two vertices of joined precisely when their labels meet, when considered as lines of PG(3,2). This displays the graph isomorphism of I' and I'* discussed at the beginning of the note. We observe that PG(3,2), or more properly the incidence structure 9, can be recovered from r. By 1.3, the maximal cliques of r are in one-to-one correspondence with the thirty distinct PG(2,2) contained in (:). Furthermore 2.2(2) and 2.2(3) show that the relation of intersecting in precisely one vertex is an equivalence relation on the maximal cliques of r having two equivalence classes each of size 15. Let these two classes be denoted fi* and X*. We form an incidence structure 9* on #z*U V ( r ) U X* where V(1') is the vertex set of 1'. A member of +* is incident with a member of %* if the two cliques concerned have non-trivial intersection. A member of V(T) is incident with each member of ,jt* and X* which contains it. N o other incidences occur. From the discussion of Section 2, it is clear that the two incidence structures 9 and 9" are isomorphic. We now calculate the automorphism group of the graph r, a u t( r ) . From our description of the incidence structure 9*, it is clear that any automorphism of I' induces either an automorphism or duality of 9 (isomorphic to .a*). Indeed, by our construction it is clear that G = S , acts naturally on X and inducing automorphisms of f.The elements of K = A, as before induce automorphisms of .a and those of G - K induce dualities of 9.As aut(9) = aut(PG(3,2))= GL(4,2), we have
r
(T)
3.1. Aut(f') has a subgroup of index 2 which is a subgroup of GL(4,2). We now define yet another graph. Let Y = {m} U X. The graph A has (;): = 35 vertices, each vertex labelled with 2 disjoint subsets of Y of size 4. Suppose that a
On identifying P G ( 3 . 2 )and the complete 3-design on 7 p i n t s
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vertex of A has label {A,B} with A U B = Y and ( A (= IBI = 4 and that a second vertex has label {C, D}. Then the two vertices are joined in A precisely when \ An
CI
=
I A no1= IBnCI = IBnDl=2.
It is clear that, by its natural action on Y and so on the labels of A, S, acts as an automorphism group of A. We define a map C$ from the vertices of A to those of r. If a vertex x of A has label {A,B} with A U B = Y and IAl= IBI = 4 then the vertex C$(x)of will be that vertex with label (a, b, c) where either {m,a, b, c} = A or {z, a, b, c} = B. We can now quickly verify that C#J is in fact an isomorphism of the two graphs A and 1’. Now S,=saut(A)=aut(r); and by 1.2 and 3.1, ( a u t ( r ) l s 2IGL(4,2)1=8!. Therefore another application of 3.1 gives
r
3.2. aut(l-) = S,. As S8 has a unique subgroup of index 2, we have an immediate corollary to 3.1 and 3.2.
3.3. A, = GL(4,2). 4. Resolutions of PG(3,2) A parallel class of PG(3,2) is a subset S of the lines of PG(3,2) such that each point is incident with exactly one line of S. Note that 15.1 = 5 and that by duality each hyperplane is incident with exactly one line of S also. A Kirkman parallelism
of PG(3,2) is a partition of the lines of PG(3,2) into 7 disjoint parallel classes. Such parallelisms provide solutions to the famous “problem of the fifteen schoolgirls” originally proposed by the Rev. T.P. Kirkman in 1850 ( [ 5 ] . A solution was given by Cayley, [2].) In this section we use our identification of the lines of PG(3,2) with the triples of ( f )to give a description of all parallel classes and all Kirkman parallelisms of PG(3,2). The five lines of a parallel class of PG(3,2) correspond to five triples of ):( with pairwise intersections never of cardinality 1.
4.1. Let S be a set of 5 triples from (T) such that x, y E S and x f y implies I x n y J ~ { 0 , 2 }Then . we have one of (1) S is all members of (q) containing a given pair {i, j } s X , or (2) S is all members either equal to or disjoint from a given member (a, b, c) of
(3. Proof. If all intersections of pairwise distinct lines in S have size 2, we must have (1). If S contains a disjoint pair of triples then all further members of S are
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disjoint from one member of the pair and have intersection of size 2 with the other. This leads to (2). Therefore we have exactly + 35 = 56 distinct parallel classes of PG(3,2), and under the action of A, these fall into two orbits of lengths 21 and 35. Of more relevance is the action of aut(PG(3,2))- A, on the parallel classes. Let the parallel class of 4.1(1) be denoted by the symbol (a,i, j ) , and let that of 4.1(2) be denoted by the symbol (a, b, c). Now the natural action of A, acting on Y = {a} U X induces action on PG(3,2) by way of the graph isomorphism 4 constructed in Section 3. Let be the associated map from pairs of disjoint subsets of given by cardinality 4 from Y into
(z)
4 (t)
where either {m, a, 6, c } = A or {m,a, b, c} = B. It can now be checked that for any element g E A, acting on Y, we have the parallel class equality ( a g ,be, cg) = ( a , b, c)&-'g&.
(This is in fact valid even for dualities from s8).A particular consequence of this is that under the action of A, = GL(4,2) the 56 parallel classes of PG(3,2) form a single orbit. Using the descriptions of 4.1, it is clear that the two parallel classes (a, b, c ) and (d, e, f) will have no common lines if and only if /{a,b, c } n { d , e, f}l = 1. Therefore the symbols for the classes of a Kirkman parallelism of PG(3,2) form a set of 7 subsets of Y with cardinality 3 with all pairwise intersections of cardinality 1. By 1.3 the symbols are therefore the lines of a projective plane PG(2,2) on 7 of the points of Y. Conversely, if we take the lines of any projective plane whose point set is contained in Y, we have the set of symbols for a Kirkman parallelism of PG(3,2). The remarks of the previous paragraph show that GL(4,2) permutes the parallelisms precisely as A, permutes the PG(2,2) composed of symbols. As before, since aut(PG(2,2))= GL(3,2)6 A,, there are 8!/168 = 240 distinct Kirkman parallelisms of PG(3,2) which under the action of GL(4,2) are permuted in two orbits of length 120. We summarize our results. 4.2. (1) PG(3,2) has 56 distinct parallel classes, all equivalent under the action of GL(4,2) = aut(PG(3,2)). (2) PG(3,2) has 240 distinct Kirkman parallelisms which fall into two dual orbits of length 120 under the action of GL(4,2).
From the construction, it is clear that the stabilizer of a given Kirkman parallelism in GL(4,2) is isomorphic to GL(3, Also, the stabilizer of a parallel class is isomorphic to a subgroup of index 2 in S3x Ss.
T).
On identifying P G ( 3 , 2 )and the complete 3-design on 7 points
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5. Planar resolutions for multiples of ($) Let A(): denote the collection of the 35 3-subsets of X,each subset repeated A times. Thus we are interested in the complete 3-design with parameters 3(7,3, A). In [l], Brouwer has studied, for various A, partitions of):(A into sets of 7 triples, each set constituting the lines of a PG(2,2). We call such a partition a planar resolution of .):(A So for instance, 2.2(1) shows that the 30 distinct PG(2,2) from (f)form a planar resolution of 6 ( 3 . It was shown by Cayley ([2]) that there is no planar resolution of (q) itself (see 5.1). Brouwer proves that planar exist for all integral A at least 2. In view of the observation resolutions of A)(: above regarding the case A = 6, it suffices to give examples with A = 2 and A = 3. These contain, respectively, 10 and 15 copies of PG(2,2). In Section 5 we use our identification of the points and hyperplanes of PG(3,2) with the subsets of (q) forming PG(2,2) to reprove Brouwer’s results and in some cases answer questions which he left open.
5.1. If A,, A,, and A, are three distinct PG(2,2) from ($) then, for some pair i and j with 1s i <j s 3, Ai nA, contains at least one line. In particular there is no planar resolution of
(T).
Proof. This is a direct consequence of 2.2(2).
5.2. No PG(2,2) can be repeated in a planar resolution of A():
if A = 2 or 3.
Proof. For A = 2 this is clear by an argument similar to that of 5.1. For A = 3, we can suppose (if necessary by applying a duality of PG(3,2)) that a PG(2,2) which is repeated twice in the resolution is a point Y of PG(3,2). As before it cannot be repeated three times. As the resolution has 15 PG(2,2), it is easy to see that of the seven hyperplanes incident to Y,at most one appears in the resolution. This quickly leads to a contradiction. A helpful observation is one already used in the proof of 5.2. Namely, if we as a collection of points and hyperlanes of think of a planar resolution of ):(A PG(3,2), then its images under any automorphism of duality of PG(3,2) will also be a planar resolution of A(:). We first consider planar resolutions of 3 ( 3 . Such a resolution has been given by Lindner and Rosa [6], and Brouwer [l] observes that there are at least three isomorphism classes (under S,) of planar resolutions of 3($). In fact, there are exactly four. Note that a planar resolution of 3(:) contains 15 of the 30 distinct PG(2,2) from (T), hence those PG(2,2) not contained in the resolution themselves form a second complementary planar resolution of 3 ( 3 . In particular, in finding all planar resolutions of 3(:) we need only find those which have more points than the hyperplanes, the rest coming from complementation.
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Let d be a planar resolution of 3 ( 3 . Suppose that under our correspondence x members of d are points of PG(3,2) and y are hyperplanes. Then we have x+y=15
and
(3.(3+3f=(3
*
35.
Here the first equation counts the members of d. The second equation counts pairs of duplicate triples occurring in d, where f denotes the number of incident point-hyperplane pairs contained in d. The only solutions in non-negative integers for x. y, f subject to x 2 y are (x, y, f ) = (15,0, O), (12, 3, 12), and (9,6, 18). One orbit of S, acting on all planar resolutions of 3 ( 3 is that which contains only two resolutions - one consisting of all points of PG(3,2) and the other consisting of all hyperplanes of PG(3,2) (under our correspondence). This is the resolution found by Lindner and Rosa [6]. We now describe all other planar resolutions of 3 ( 3 in terms of the corresponding configurations in PG(3,2). 5.3. Let d be a set of 15 points and hyperplanes from PG(3,2) consisting of 12 points and 3 hyperplanes. Then d corresponds to a planar resolution of 3 ( 3 if and only if d consists of all points of PG(3,2) not incident to a given line 8 and all three hyperplanes incident to t'.
Proof. We first observe that a configuration from PG(3,2) consisting of all points not incident to given line and all hyperplanes incident to the line furnishes a planar resolution of 3 ( 3 . Conversely, if 1 is a planar resolution of 3 ( 3 consisting of the twelve points not incident to a given line t' plus 3 additional hyperplanes, these hyperplanes must be those incident to t'. Thus it suffices to show that if a planar resolution d of 3 ( 3 contains exactly 12 points, then the points are those not incident to a given line of PG(3,2).Assume that the three deleted points p, q and r do not lie on a common line. Each pair of points from {p, q, r} is incident with a unique tine of PG(3,2) and each line as determined is incident with only one point of d. Therefore each of these lines must be incident with exactly 2 hyperplanes in d.This requires at least four hyperplanes, a contradiction which proves 5.3. There are 35 planar resolutions of 3(:) of the type described in 5.3, and in addition there are 35 complementary resolutions. As A, is transitive on the lines of PG(3,2). there 70 planar resolutions of 3(:) form a single orbit under t h e action of S,. 5.4. Let d be a set of 15 points and hyperplanes from PG(3,2) containing 9 points and 6 hyperplanes. Then d corresponds to a planar resolution of 3 ( 3 if and only if the points of PG(3,2) not in d are those incident with two disjoint lines t',and 8, and the hyperplanes of PG(3,2) in d are those incident to 8 , and t2.
On identifying P G ( 3 , 2 ) and the complete 3-design on 7 points
139
Proof. We first observe that if SB contains the 9 points not incident to two given disjoint lines, then for d to correspond to a planar resolution of 3 ( 3 the six hyperplanes of d must be those incident to the two lines. Furthermore, such an d does produce a planar resolution of 3 ( 3 . Thus we need only prove that a planar resolution d of 3(:) containing 9 points and 6 hyperplanes has its points those not incident to two disjoint lines. Let p, q and r be three points not in d. We claim that for some line e incident with two of p, q and r the third point incident to 4 is also not in d. Assume otherwise that p, q, r are not collinear and so are incident with a unique common plane Q. Our assumption implies there is a line incident with Q all of whose points are in d. Thus Q is not in d and the hyperplanes of d are precisely those 6 incident with two of p, q and r. Now as Q is incident with at most 4 points of d, there are at least 21 incident point-hyperplane pairs in d.The contradiction verifies our claim. As the claim is valid for all triples of points not in d, we find that the points not in d either are those incident with two disjoint lines or are all incident to a common hyperplane. In this second case, counting incident point-hyperplane pairs in d again produces a contradiction and completes the proof of 5.4. A, has two orbits on disjoint pairs of lines from PG(3,2) corresponding to pairs of triples from (T)with intersections of size 0 and 2. The orbits have lengths 70 and 210 respectively. As these numbers are distinct, a planar resolution of the type described in 5.4 must be in the same orbit under S, as its complement. Therefore 5.5. There are exactly 632 distinct planar resolutions of 3 ( 3 falling into four orbits under S, of lengths 2, 70, 140 and 420.
(T)
5.6. There are exactly 316 different partitions of the 30 distinct PG(2,2) from into two sets of 15, each set providing a planar resolution of 3 ( 3 . The 316 fall into four orbits under S, of lengths 1, 35, 70 and 210.
We now consider planar resolutions of 2 ( 3 . These have been discussed rather thoroughly by Brouwer in [l], so we do not give all our proofs in detail. For our purposes, it is convenient to have the concept of an ovoid of PG(3,2). An ovoid is a set of 5 points of PG(3,2) no 4 of which are incident to a common hyperplane. Ovoids are easy to construct - any three points not incident to a common line can be extended to an ovoid in exactly 4 ways. From the construction, it is clear that GL(4,2) is transitive on the ovoids of PG(3,2), there being 168 distinct ovoids. (The global stabilizer of an ovoid is isomorphic to S5 and is generated by transvections.) Using the results of Section 2 we can check that the ovoids break into two orbits of length 42 and 126 under the action of A,. It is easy to see that for each point p of an ovoid there is a unique hyperplane tangent to the ovoid at p, that is, incident to p but to no other point of the ovoid. These 5 tangent hyperplanes in fact form a dual ovoid of PG(3,2).
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5.7. Let d be a set of 10 distinct points and hyperplanes of PG(3,2). Then d corresponds to a planar resolution of 2 ( 3 if and only if d is the set of points and tangent hyperplanes of some ovoid of PG(3,2).
Proof. Suppose ‘t is a line of PG(3,2) incident with no point of an ovoid 6. As no 4 points of 6 are incident to a common hyperplane, one of the hyperplanes containing t‘ is incident with 3 points of 6 and the other two hyperplanes containing t‘ are tangent to 6. With this in mind, it is easy to see that the points of an ovoid and its tangent hyperplanes d o correspond to a planar resolution of 2 ( 3 as desired. Now assume that d is a collection of 10 points and hyperplanes of PG(3,2) which corresponds to a planar resolution of 2(3. Let x be points of PG(3,2) and y be hyperplanes. As before, we have x+y=lO
and
(2”) + (2’) +
3f=
(;)
*
35,
where f is the number of incident point-hyperplane pairs in d. The only non-negative integral solutions are (x, y, f ) = (2,8,2), (8,2,2) and (5,5,5). Any 2 distinct hyperplanes of PG(3,2) are incident with all but 4 points of PG(3,2). This and a dual observation immediately rule out the possibility of configurations d with (x, y,f)=(8,2,2) or (2,8,2). Therefore we must have (x, y,f)=(5,5,5). Clearly no three points in d can be incident to a common line of PG(3,2), therefore if 4 points of d are incident to a common hyperplane Q they must be those not incident to some given line contained in Q. But in that case, at least 3 hyperplanes of d would be incident to 2 or more points of d, contradicting f = 5 . Thus the points of d are those of an ovoid in PG(3,2). It now easily follows that the hyperplanes of d must be those tangent to the ovoid. This gives 5.7 and 5.8. (Brouwer, [l]). There are 168 planar resolutions of 2 ( 3 which fall into orbits of length 42 and 126 under the action of S,. Brouwer [l] describes all ways of partitioning the 30 distinct PG(2,2) from (t) into three disjoint planar resolution of 2(:). In PG(3,2) such a partition corresponds to 3 pairwise disjoint ovoids. Considering the PG(2,2) from ):( we see that we need only find two disjoint ovoids to insure that the remaining 5 points also form an ovoid. We have seen above that any hyperplane of PG(3,2) is either tangent to a given ovoid or is incident to three points of the ovoid, these three points of course being incident to no common line (they form a triangle). Thus for any given hyperplane Q and any given set of three disjoint ovoids, one of the ovoids has Q as a tangent hyperplane while the other two ovoids are incident to two disjoint triangles of points from Q. There are 42 pairs of disjoint triangles in the hyperplane Q, and each pair can be completed to two disjoint ovoids in 8 different ways. Hence
On identifying PG(3.2) and the complete 3-design on 7 points
141
5.9. (Brouwer, [l]).There are 336 different partitions of the 30 distinct PG(2,2) from (t)into 3 sets of 10, each of which is a planar resolution of 2(?).
Brouwer shows that the 336 partitions come in two orbits of lengths 126 and 210 under S,. Again, this can be checked by using the action of A, on PG(3,2) given in Section 2.
Note added in proof A similar identification, due to Gleason, may be found in Wagner, Math. Zeit. 76 (1961) 424.
References [l] A.E. Brouwer, A note on the covering of all triples on seven points with Steiner triple systems, Mathematish Centrum ZN 63/67, 1976. [2] A. Cayley, On the triadic arrangements of seven and fifteen things, London, Edinburgh, and Dublin Philos. Mag. and J. Sci., (3) 37 (1850) 50-53. (Collected Mathematical Papers, Vol. 1, 481-484). [3] P. Dembowski, Finite Geometries, Springer-Verlag, Berlin-Heidelberg-New York, (1968). [4] M. Hall, Jr., Combinatorial Theory, Blaisdell, Waltham-Toronto-London, (1967). [5] T.P. Kirkman, Query, Lady’s and Gentlemen’s Diary (1850) 48. [6] C.C. Lindner and A. Rosa, Construction of large sets of almost disjoint Steiner triple systems, Canad. J. Math. 27 (1975) 256-260.
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Annals of Discrete Mathematics 7 (1980) 143-150 @ North-Holland Publishing Company
A SURVEY ON RESOLVABLE QUADRUPLE SYSTEMS Alan HARTMAN Department of Mathematics, Uniuersify of Newcastle, New South Wales, 2308, Australia A Steiner system ( X . p), denoted $(I, k, 0). is a set X of poinfs, of cardinality u, and a collection p of k-subsets of X called blocks, with the property that every 1-subset of X is contained in precisely A blocks. A quadruple system is a Steiner system S,(3,4, 0). A triple (X,p . y ) is called an (s, p)-resoloable system if, for some s < I , it is a partition of an SA(r,k, c ) system (X, p ) into subsystems ( X , y , ) , each of which is an S,(s, k. v ) system, such that y = y , I y z I . . I yc is a partition of 0. A system is doubly resolvable if it is resolvable and each ( X , yi) is also resolvable. This article surveys the work done on the existence of (s, @)-resolvable and doubly-resolvable quadruple systems for (s, /A) = (2, 1). ( 2 , 3 ) and ( 1 , 1).
Introduction and terminology A Steiner system (X, p ) , denoted SA(r,k, u ) , is a set X of points, of cardinality u, and a collection p of k-subsets of X , called blocks, with the property that every 1-subset of X is contained in precisely A blocks. The number of points, u, is the order of the system. Where blocks are listed explicitly square brackets will be used. A quadruple system is a Steiner system S1(3,4, u ) . A system Sl(l,k, u ) is simply a partition of X into k-subsets, and is called a parallel class. A triple (X, p, y ) is called an (s, p)-resolvable system if, for some s < 1, it is a partition of an SA(f.k, u ) system (X, p ) into subsystems ( X , y,), each of which is an S,(s, k, u ) system, such that y = y1 1 y2 1 . . * 1 yc is a partition of p . The 7,’s are called resolution classes. A resolvable system ( X , p, y) is doubly resolvable if each (X, y I ) is resolvable. The existence of resolvable Steiner systems was first raised in 1847 by Kirkman [ 101 in his well-known “schoolgirl” problem. Kirkman asked for the determination of necessary and sufficient conditions on u for the existence of (1, 1)resolvable S,(2,3, u ) systems. This task was not completed until 1961 when Ray-Chaudhun and Wilson [12] showed that these systems exist if and only if u = 3 (mod 6). The literature on existence questions for resolvable Steiner systems is quite extensive: a bibliography is given in [7]. Necessary and sufficient conditions for the existence of quadruple systems were established only in 1960 by Hanani [5],when he showed that a quadruple system of order u exists if and only if u = 2 or 4 (mod 6). Although (1, 1)-resolvable quadruple systems of orders u=2”’ were known to Kirkman, the existence problems for resolvable quadruple systems have been tackled seriously only since 1976. This article is an attempt to summarize the research on these problems by 143
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144
Baker [ 11, Booth [2], Greenwell and Lindner [4] and the author [6,8,9]. These papers construct (s, p)-resolvable and doubly resolvable quadruple systems for which (s, p ) has the values (2, l),(2,3) and (1, 1).Baker [l]has constructed (2, 1) (1, 1)-doubly resolvable systems of all orders u =4". In [9], the author has constructed (2, 3) (1, 1)-doubly resolvable systems of orders u = 8, 20, 32, 44, 68, 80 and 104, and has also given a (2,3)-resolvable system of order u = 128. The other papers cited construct (1, 1)-resolvable quadruple systems. At this point we note a result of a more general nature on resolvable quadruple systems. Lindner [ 111 has shown that every quadruple system can be embedded in a (1, 1)-resolvable quadruple system.
1. Algebraic construction In this section we describe constructions due to Baker [l] and the author [8,9] which have an algebraic flavour. The quadruple system (X, p ) is constructed using an algebraic structure on the set X , and the algebraic properties of the system are used to facilitate resolution.
1.1. Afine consfrucfion Kirkman [lo] showed that for any n > 1, the planes of the affine space of dimension n over GF(2) form the blocks of a quadruple system of order u = 2". These systems are (1, 1)-resolvable by taking each parallel class to be a two dimensional subspace and all of its translates. Baker [ 11, in 1976, gave a slightly different presentation of these affine spaces for even n, and demonstrated a partition of their planes into S,(2,4,2") systems each of which is (1,1)-resolvable. This construction will now be described. Let X = GF(22m-1) x GF(2), then X can be considered either as the set of points (a, i) with a E G F ( ~ ~ " - ' )i,E ( 0 , 1) or as the set of vectors of length 2m over GF(2). Let p be the set of planes in the vector space over X. Writing the block BE^ as
B =[(a,, i,)(a2, i * k , iAa4, i4)1, define
Yb
by
where 6 is the Kronecker symbol and b E GF(22"-') \ (0). Some technical lemmas are needed to show that: (i) y = {yb :b E GF(22m-1)\ (0))is a partition of p ; (ii) ( X , yb) is an S,(2,4, Pm) system for each b; and (iii) if B E y b then B + ( a , i) E yb for all a, b and i.
Resolvable quadruple sysrems
145
The reader is referred to the original paper [ 11for proofs of these results. The first two points show that we have indeed constructed a (2, 1)-resolvable quadruple system, and the third shows that the partition y refines to the usual (1, 1)resolution of the affine space. 1.2. Projective construction In the 1930’s Carmichael [3] constructed two families of quadruple systems with highly transitive automorphism groups, on the projective line X = GF(q)U {m}.The first of these constructions has q = 3” and p is the orbit of the quadruple [m,0,1,2] under the action of the group GL(2,3“) defined by GL(2,3”) = {x
I+
(ax + b)/(cx + d ) :a, b, c, d E GF(3”), ad - b c f 0).
No work has yet been done on the resolution of these systems, although the field seems promising. The second construction, which is described below, has been investigated and shown to generate quadruple systems which are “easily” resolvable by computer search. Let q = 7(mod 12) be a prime power and let g be a primitive cube root of unity in GF(q). As before take X = GF(q) U {m},but now take 0 to be the orbit of the quadruple [m,0,1, g + 11 under the action of the group SL(2, q ) defined by
SL(2, q) = {x
-
(ax + b)/(cx+ d ) :a, b, c, d E GF(q), ad - bc = 1).
Carmichael [3] showed that (X, p ) is a quadruple system. Let e(m, c) denote the transformations of X defined by
Let w be a generator of the multiplicative group on GF(q) \ {0}, and define the following subsets of SL(2,q): Q={e(m, c ) : c ~ G F ( q ) m , =w2’,Osi
G = {e(l,01, e(g, o), e ( g 2 , o)}, T = {O(l, c ) : c E GF(q)} and
P = {e(m, 0) : m = w2’, 0 s i < i ( q
- 1)).
If A €3 denotes { f o g : f E A , g E B}, then it is clear that P o T o G = Q. In [8] the author has determined the structure of the Q, P, T and G orbits of quadruples in p, and shown that if a suitable set a of distinct representatives of the Q-orbits exists then the system is either (1, 1) or (2,3)-resolvable or doubly resolvable with those parameters. For (1, 1)-resolvability the “suitable” sets a have the property that G ( a )is a parallel class S l ( l ,4, q + 1).For (2, 3)-resolvability the suitable sets are such that ( T o G) (a)is an S,(2,4, q + 1) system, and for double-resolvability both these properties are satisfied. The technique described above applies to any 0
146
A. Harfman
quadruple system on X = GF(q)U {m} with 0 as a subgroup of its automorphism group. In [8] suitable sets a are given for (1, 1)-resolvability in the cases where q 7 (mod 12) is a prime power and 7 S q G 379. In [9] suitable sets and appropriate quadruple systems are given for (2,3)-resolvability where 7 S q S 127 and for double resolvability when 7 s q S 103. The existence of these sets for all prime powers q = 7 (mod 12) remains an open question. The technique described above is also useful for generating solutions to another resolvable design problem. Two (1, 1)-resolutions ( X , p, y) and (X, p, 6) of a quadruple system ( X , p ) are orthogonal if \yin s 1 for all i, j . It is conjectured that a set of $(u-2) mutually orthogonal resolutions exist whenever u = 8 (mod 12j, and this has been verified for u s 4 4 in [9].
2. Inductive constructions Throughout this section we will be concerned with (1, 1)-resolvable quadruple systems and accordingly we will omit the prefix (1, 1). Resolvable quadruple systems exist only if u = 4 or 8 (mod 12) and many authors have conjectured that this condition is also sufficient. In support of this conjecture we note that the smallest resolvable quadruple system whose existence is not yet demonstrated is one of order 76, and only twenty of the sixty-seven possible orders u 5 400 are in question. A series of inductive constructions is necessary to construct resolvable quadruple systems of all admissible orders. Two such constructions are known; a doubling construction used by Booth [2] and Greenwell and Lindner [4] to construct resolvable systems of order 20 and 28 respectively, and a tripling construction due to the author [6]. 2.1. Doubling consfrwcrions The doubling construction given below is the one used by Booth [2] and Greenwell and Lindner [4] to construct the first known resolvable quadruple systems whose order is not a power of 2. It is a generalization of the construction used by Hanani IS] and known to others before him. We begin with some definitions. A partial parallel class a in a quadruple system (X, p ) is a set of blocks containing each element of X at most once. A complement a* of a partial parallel class a is a set of 2-subsets of X such that a U a* is a partition of X. Note that for a given cx there may be more than one possible complement a*. A one-factor 4 of the complete graph on X is a set of 2-subsets of X which partitions X. A one-factorization of the complete graph on X is a partition @ = 4, I 421 . . 1 of the set of all 2-subsets of X into one-factors. As well as numbering the one factors in a one-factorization 1, 2, . . . , u - 1 we shall also number the 2-subsets in each one factor 1, 2 , . . . , ; u .
+"-,
Resolvable quadruple sysferns
147
Theorem 2.1 Let ( X , p ) be a quadruple system of order u whose block set p can be partitioned into partial parallel classes al,ciz, . . . , such that there exist complements a:, a:, . . . ,with the properties: (i) a: nay = @ for all i f j ; (ii) u a ? can be partitioned into m one-factors & 1 42I - * I&, of the complete graph on X ; and (iii) the set { & l , & 2 , . . . , &,} extends to a one-factorization @ = &u-l of the complete graph on X . 1&, * . Then there exists a resolvable quadruple system of order 2u. Proof. We construct a resolvable quadruple system of order 2 u as follows. The point set X is “doubled” by taking XI= X X ( 0 , l}, and writing xi for each ( x , j) E X ‘ . The blocks and parallel classes on X’ are constructed as follows. First, for each partial parallel class aiand its complement a:, we “double” ai and a? to form a parallel class y ; on XI. Specifically, y; comprises [x,, y o , zo, to] and [ x , , y l , z , , t , ] corresponding to each [ x , y , z, t ] ai, ~ and [ x o , yo, x , , y l ] corresponding to each [ x , y ] a:.~ Let cr be the cyclic permutation (12 $u). Each remaining parallel class y L (where 1S i < u and 0 < k <$I) comprises all blocks of the form [x,, y o , zl, r,] where [ x , y ] and [ z , t ] are respectively the jth and a k ( j ) t h members of the one-factor &. Finally, delete the classes ylo with l S i S m ; their blocks have already been assigned to parallel classes as they arise from doubling an [ x , y ] in some a:. 0 Note that this theorem implies that a resolvable system of order 2u can be constructed from a resolvable system of order u, by taking the ai as parallel classes with empty complements and an arbitrary one-factorization @.
2.2 Tripling constructions In 1978 the author [ 6 ] published the result that if a resolvable quadruple system of order u = 8 (mod 12) exists then a resolvable quadruple system of order 3u-8 exists. This section contains a discussion of that result. A resolvable subsystem of a resolvable quadruple system (X, p, y) is a resolvable quadruple system (S, ps, y s ) with S G X, ps G p and for each ( y s ) iE ys there exists a yj E y such that yi. Note that every resolvable quadruple system contains a resolvable subsystem of order 4. A resolvable subsystem ( S , ps, y s ) induces natural partitions on X, p and the members of y in the following way. Let 7 be the set of partial parallel classes which extend the parallel classes in ys, and let 7 be the set of parallel classes on X containing no parallel class - from ys. Let p be the set of blocks contained in elements of 7, and define 6 similarly. Finally let N = X \ S, and we can now write the resolvable system (X, p, y ) with resolvable subsystem (S, ps, ys) as
( S u N, P s u P u
F,Ys u r u
9
9
148
A. Hartman
Fig. 1
where all the unions are disjoint. A diagrammatic representation of such a system is given in Fig. 1. Each cell of the matrix contains a single block and each row of the matrix is a parallel class. The tripling construction in [6] has the following form. Let (SUN, P s U p U i , ysUYUT) be a resolvable quadruple system of order n + s = 8 (mod 12) with a resolvable subsystem of order s = 4. The resolvable system ( X ’ ,p’, y ’ ) of order 3 n + s is defined as follows. Let X‘= S U N x ( 0 , 1,2}, and again we write n, for (n,j ) E N x ( 0 , 1,2}. The block set p‘ is defined as the disjoint union of ten sets of blocks denoted by ps, e, and 6. The sets comprise blocks of the form [x,, y,, z,, r,] corresponding to each [x, y, z, t ] ~ p and , sirnilarly for the blocks in although elements of S are not subscripted. Blocks in 8 have the form [x, yo, zl, r2] where x E S and y, z , t E N ; blocks in I#I have the form [x,, y,, z,, r k ] where x, y, z , r E N and {i, j , k } = { O , I , 2); and blocks in 6 have the form [x,, y,, z,, f,] where x, y, z, f e N and i f j. The parallel classes y’ are arranged according to the diagram of Fig. 2. The reader is directed to [6] for an explicit construction of the blocks in 0, 4 and 5, and their arrangement into parallel classes. If n = 12k + 4 then different constructions of the blocks in 6 are needed when k is even or odd. Analogous constructions in the five cases
Po,
PI. P2.&, gl,g2,
fit,
s=4, n = 1 2 , s = 8,
n = 12k or 12k + 8 ,
s = 16 or 20,
n = 12k + 8
149
Resoloable quadruple sysiems
4+++;~-:---+-;+
-BOYD
El71
6
Fig. 2.
8272
150
A. Hartman
could lead to a complete solution of the existence problem for resolvable quadruple systems. However if the complexity of the known construction with s = 4, n = 12k + 4 is any guide then the project is a daunting one. Another avenue of attack is provided by the doubling construction of Theorem 2.1. If one could show that for every u = 2 or 4 (mod 6) there was a quadruple system of order u satisfying the conditions of Theorem 2.1, then the problem would be solved. In view of the limited success achieved so far, using that avenue, this too would seem to be a very difficult task.
References [ l ] R.D. Baker. Partitioning the planes of AG,,,(2) into 2-designs, Discrete Math. 15 (1976) 205-2 1 1. [2] T.R.Booth, A resolvable quadruple system of order 20, Ars Combinatoria 5 (1978) 121-125. [3] R.D. Carmichael. Introduction to the Theory of Groups of Finite Order (Ginn, Boston, MA, 1937: reprinted by Dover. New York, 1956). [J] D.L. Greenwell and C.C. Lindner. Some remarks on resolvable quadruple systems, Ars Combinatoria 6 (1978) 215-221. [S] H. Hanani. On quadruple systems, Canad. J . Math. 12 (1960) 145-157. (61 A. Hartman. Parallelism of Steiner quadruple systems, Ars Combinatoria 6 (1978) 27-37. [7] A . Hartman. A survey of the existence problem for resolvable designs. t o appear in Proceedings of the First Franco SEA Math. Conf. Workshop on Combinatorics. [8] A. Hartman. Resolvable Steiner quadruple systems, to appear in Ars Combinatoria. [9] A . Hartman. Doubly and orthogonally resolvable quadruple systems, to appear in Proceedings of the 7th Australian Combinatorics Conference. [to] T.P. Kirkman. On a problem in combinations, Cambridge and Dublin Math. J . 2 (18471 I9 1-204. [ I I] C.C. Lindner. Every Steiner quadruple system can be embedded in a resolvable Steiner quadruple system. Ars Combinatoria 3 (1977) 75-88. [ 121 D.K. Ray-Chaudhuri and R.M. Wilson. Solution of Kirkman’s schoolgirl problem. Proceedings of Symposia in Pure Mathematics, Vol. 19. in: T.S. Motzkin, ed.. Combinatorics (Amer. Math. Soc.. Providence. RI.. 197 I ) 187-204.
Annals of Discrete Mathematics 7 (1980) 151-173 @ North-Holland Publishing Company
PROJECTIVE EMBEDDINGS OF SMALL “STEINER TRIPLE SYSTEMS” Monique LIMBOS Deparrernenr de Mathirnafique, Unioersiri Libre de Bruxelles, Boulevard d u Triornphe, 10.50 Bruxelles, Belgium
Received 15 November 1979 A Steiner triple system S is embeddable in a finite Desarguesian projective plane P i f rhere exists n subser S‘ of P such that S’ provided with the restrictions of the lines of P is a Steiner triple system isomorphic to S. I n this note we show that among the two systems of cardinality 13 and the eighty systems of cardinality 15, only one is embeddable in a finite Desarguesian projective plane, namely the 3-dimensional projective space over GF(2).
1. Introduction A Steiner triple system S(2,3, u ) is a set S of cardinality u whose elements are called points, provided with a collection of distinguished subsets of cardinality 3, called blocks, such that any two points are incident with one and only one block. We will say that a Steiner triple system S is ernbeddable in a finite Desarguesian projective plane P if there exists a subset S’ of P such that S’, provided with the restrictions of the lines of P, is a Steiner triple system isomorphic to S. From now o n , when we say “embeddable” (or “embedded”) this will mean “embeddablc” (or “embedded”) in a finite Desarguesian projective plane”. It is well known [2] that t h e finite dimensional affine spaces over GF(3) are embeddable. A finite dimensional projective space of GF(q), P,(q), is trivially embeddable in P , , ( q r ) .If there exists in P,,(q‘) a point p which lies in no plane of P,(q) it is possible to project P n ( q ) from p into an hyperplane Pn-l(q)in such a way that three non collinear points remain non collinear. An easy but tedious counting argument shows that if r is sufficiently large such a point does exist. By itcration, P,(q) is embeddable in a plane P,(q”’) for a certain m. Therefore the finite dimensional projective spaces over GF(2) are embeddable systems. In particular, the unique S(2,3,7), which is the projective plane over GF(2), is embeddable (as a subplane) in every plane over a field of characteristic 2, and the unique S ( 2 , 3 , 9 ) , which is the affine plane over GF(3), is embeddable (as a subplane) in every plane over a field of characteristic 3, and is also embeddable (but not as a subplane) in every plane over a field F in which the polynomial x z + x + 1 is reducible 131. 151
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M. Limbos
As far as we know no other examples of embeddable Steiner triple systems have been found. In this note we show that, among the systems of cardinality 13 and 15, only one is embeddable, namely the 3-dimensional projective space over GF(2).
2. The two S(2,3,13)
It is well known that there are exactly two non isomorphic S(2, 3, 13) which we shall denote by S , and S,.
2.1. S, can be described as follows: -the
points of S , are the elements of Z,3, of S , are the subsets { 1 + i, 2 + i, 5 + i} and { 1+ i, 6 + i. 8 + i} where
- the blocks
i
EZ,,.
The automorphism group of S,, of order 39, is generated by the permutations i - + i + l and i-+3i. Note that S, contains the complete quadrangle shown in Fig. 1. If S , is embeddablc, we may assume, without loss of generality, that the blocks { 1.2. S}, { 2 , 3 , 6 } ,{S,6,9} and { 1,3,9} are subsets of the lines of equations x = 0, y = 0. z = 0 and x + y + z = 0 respectively. We deduce immediately the coordinates of the following points: 2 = ( 0 , 0 , 1).
S = ( O , l.O),
6 = ( 1 , O, O),
3 ~ ( 1 . 0 -, l ) ,
l ~ ( 0 1,-1). ,
9~(1,-1,0)
The block { 1,6,8}is a subset of the line y + z = 0 and so 8 must have coordinates (a,l . - l ) with a Z O . Similarly the blocks { 3 . 5, 1 I} and {2,9,7}are contained in the lines x + z = 0 and x + y = 0 , so that 1 1 = ( - l , p , 1) with p f 0 , and 7 = ( 1 , - 1 , y ) with y f O . The point 4 belongs to the blocks {3,7,4}, {8,5,4} and (9, 11,4}. Since the coordinates of t h e points 3. 7, 8. 5 , 9 and 11 are already known. we get
Fig. 1
Projectioe embeddings of small Steiner Triple Systems
153
4 = ( a ,p + a - 1, 1). Moreover a,p and y must satisfy the relation 1 - a = (1 + y ) ( 1 - a - p ) .
(1)
The automorphism i+3i, namely (0)(1,3,9)(2,6,5)(4,12, 10)(7,8,1l), permutes the blocks { 1,2,5}, {3,6,2}, {9,5, 11) cyclically and preserves the block { 1,3,9}. Therefore, by permuting cyclically the coordinates of 4 and the parameters ( a ,p, y), we get the coordinates of 12 and 10 together with the relations:
1 - p = ( 1+ a ) ( 1 - p
(2)
-y),
1 - y = (1 +@)(1-y-a).
(3)
In order to determine the coordinates of 0, we consider the blocks {0,6, ll}, {0,8,2}, (0, 7,4} and we get 0-(ap, p, 1) with a p y = 1.
(4)
The blocks {2,4, lo}, { 5 , 10, 12}, {4,6, 12}, (0, 1,4}, {0,3, 12}, {0,9, 10) imply that the parameters must also satisfy the relations: a z = a + p + y - p y - a y - a p - 1,
(5)
p z = a +p +y-
p y - a y - a p - 1,
(6)
ap - 1,
(7)
+p +y a =p +1
yz= a
py- ay-
(8)
+p-I,
p = y + 1+ y - l , y = a + 1+ a - I .
From ( 5 ) , (6) and (7) we deduce a’= p’= y2. This implies that at least two of the parameters are equal, and because the relations are symmetric with respect to the parameters, it is no loss of generality to assume that a = p. Then (by (8)) p = - 1 = a, and (by (10)) y = - 1. Therefore a p y = -1 which contradicts (4), except if the characteristic of the ground field is 2. But in this case a = p = y = 1 = -1, and so the points 7, 8 and 11 are not distinct which is again a contradiction. 2.2. S, can be constructed as follows:
-the points of S, are the points of S , , - the blocks of S, are the blocks of Sl, except the four blocks { 1, 2,5}. {4,6,9}, { 1,3,9} and {2,3,6} which are removed and replaced by {1,2,3}, {1,5,9}, {3,6,9} and {2,5,6}. Observe that i 4 3 i is again an automorphism of S,. Like in case 2.1 we may assume that:
1 -(0,0, l),
3=(1,0,0),
9=(0, l,O),
2=(1,0, -l),
5 ~ ( 0 , - 1 , l),
6~(1,-1,0).
M. Limbos
154
and, searching the coordinates of the points 7, 8 and 11, we introduce the non-zero parameters y, a, p. After analogous computations we get the following relations: 1 + a = y( 1- p ) , 1+ p
= a(1- y ) ,
1 + y = pc 1 - a ) , 1 - 0 = p(a-y),
1 - p = ycp
-
a),
I-y=a(y-P),
acp - y , p I = 0, p ( y - (Y ) y - I = 0, y(a-0)cu-l
=o,
-spy = a p + a y + py. Since a, p, y are non-zero parameters, (7) (8) and (9) imply a
=p =
y. T h e n (by
( I ) ( 2 ) and (3))we find a 2 = p 2 = y ' = - l , and (by (4) (5) and (6)) a = @ = ? =1. This is not possible except if the characteristic of the ground field is 2; in this case however the points 7. 8 and 11 are not distinct which is a contradiction. 3. The eighty S(2,3,15) In this part we use the table of Seidel and Bussemaker [4] which describes the complete list of blocks of the 80 S ( 2 . 3 , 15). With the kind permission of the authors we reproduce this table in the appendix. I t is well-known that the system No. 1. which is the 3-dimensional projective space over GF(2). is embeddable. We shall be concerned with the 79 remaining S ( 2 . 3 , IS).
3.1. If an S ( 2 . 3 , u ) containing an S ( 2 , 3 , 7 ) is embedded in a projective plane P, the subsystem is also clearly embedded in P. Thus P is necessarily of characteristic 2 and each quadrangle of P must have collinear diagonal points [ 11. So if a Steiner triple system S contains an S ( 2 , 3 , 7 ) and a quadrangle whose diagonal points do n o t form a block, S is not embeddable. With t h e aid of a computer we have found that among the eighty S ( 2 , 3 , 15). thc systems No. 2, 3, 4, 5 , 8 , 9. 10, 1 1 , 12. 13, 14, 1.5, 16, 17, 18 are of this type and s o are not embeddable. (This had already been established independently by Monique Vandensavel). We have still 64 systems to examine.
3.2. I t is well-known that every finite Desarguesian projective plane P is Pappian, that is: if a, 6, c and d. e. f are collinear points of P, then the points
155
Projective embeddings of small Steiner Triple Systems a
b
C
d
a
f
Fig. 2.
g -- ae n bd, h = af f l cd and i = b f n ce are collinear. Thus an S(2,3, u ) is not embeddable if it contains the configuration shown in Fig. 2, in which {g, h, i} is not a block. We give below the list of all S(2, 3, 15) containing at least one such configuration. For each of them we describe one “forbidden” configuration in the form of an ordered list: (n) a, b, c, d , e, f, g, h, i where n is the number of the system, and the list a , . . . , i corresponds to the above picture.
(27) 4, 9, 10, 11, 6, 5 , 2, 1, 15
(28) 2, 15. 13, 12, 6, 5 , 4, 1, 10
(32) 4, 9, 10, 11, 6, 5 , 2, 1, 15
(34) 1, 15, 14, 12, 6, 3, 7, 2, 10
(38) 1, 12, 13, 10, 8, 2, 9, 3, 14
(39) 1, 14, 15, 10, 7, 4, 6, 5 , 9
(42) 2, 15, 13, 12, 6, 3, 4, 1, 10
(45) 2, 9, 11, 10, 13, 3, 15, 1, 5
(46) 1, 13, 12, 11, 5 , 6, 4, 7, 8
(48) 1, 13, 12, 11, 5 , 6, 4, 7, 8
(49) 1, 4, 5 , 6, 3, 12, 2, 13, 10
(50) 1, 9, 8, 4, 6, 2 , 7, 3, 11
(51) 2, 7, 5 , 4, 8, 3, 10, 1, 15
(52) 2, 6, 4, 5 , 11, 3, 9, 1, 12
(53) 2, 8, 10, 5 , 13, 14, 15, 12, 6
(54) 4, 7, 12, 13, 14, 5 , 11, 1, 2
( 5 5 ) 1, 2, 3, 10, 9, 15, 8, 14, 13
(56) 2, 6, 4, 5 , 11, 3, 9, 1, 12
(57) 2, 7, 5 , 4, 8, 3, 10, 1, 15
(58) 4, 7, 12, 13, 14, 5 , 11, 1, 2
(59) 1 , 2 , 3 , 1 1 , 8 , 6 , 9 , 7 , 4
(60) 1 , 2 , 3 , 1 1 , 8 , 6 , 9 , 7 , 4
(65) 1, 14, 15, 12, 9, 2, 8, 3, 11
(66) 1, 13, 12, 3, 8, 4, 9, 5 , 14
(67) 1 , 2 , 3 , 1 2 , 8 , 6 , 9 , 7 , 4
(68) 1, 14, 15, 12, 9, 2, 8, 3, 11
(69) 1, 10, 11, 15, 13, 2, 12, 3, 8
(70) 1, 4, 5 , 6, 3, 12, 2, 13, 14
(71) 1 , 2 , 3 , 1 2 , 8 , 6 , 9 , 7 , 4
(72) 1 , 2 , 3 , 1 2 , 8 , 6 , 9 , 7 , 4
(73) 1 , 2 , 3 , 1 2 , 8 , 6 , 9 , 7 , 4
(75) 1 , 2 , 3 , 1 2 , 8 , 6 , 9 , 7 , 4
(77) 8, 5 , 13, 12, 2, 9, 10, 1, 15
(78) 6, 9, 15, 14, 10, 7, 5 , I , 3
156
M. Limbos
This list was also established with the aid of a computer. We have eliminated 34 new systems but there are still 30 systems to examine.
3.3. Suppose that an embeddable S(2,3, u ) contains the configuration shown in Fig. 3. Since the triangles dfe and gih are such that a = d e n gh, b = e f n hi and c = dfn gi are collinear, Desargues’ theorem implies that the blocks gs, fi and eh either have a common point, or are disjoint because the lines gd, fi and eh must be concurrent. a
C
Fig. 3
We give below the list of all S(2,3. 15) which contain the above configuration bur in which two of the blocks gd, fi and eh have a common point while the third block is disjoint from them. Such a system is therefore not embeddable. For each of them, we give the ordered list ( n ) a, b. c. d, e, f, g, 11, i. (*)
(2) 1. 2, 3, 5, 4, 6. 9. 8, 10
( * I ( 4 ) 1. 2, 3, S, 4, 6, 9, 8, 10
(6) 1 . 2 , 3 . 9 . 8 , 1 0 , 4 , S . 7
(*)
(3) 1 , 8, 9, 1 I , 10, 2, 13. 12, 6
(*) (5) 1, 2, 3. 5 , 4. (*) (8)
6, 13, 12, 14
I , 2, 3, 8. 9, 11, 13, 12, 14
(*I ( 9 ) 1. 2. 3. 7, 6, 4, 11. 10. 8
(*)(lo) 1, 2, 3, 5 , 4, 6, 8, 9, 11
(*)(I I ) I , 2, 3. 5 , 4. 6, 1 1 , 10. 8
(*)(12) 1, 2. 3, 5 , 4, 6 , 8, 9, 1 1
(*)(13) 1. 2, 3. 5. 4, 6. 8, 9, 1 1
(*)(14) I , 2. 3, 7 , 6 , 4 , 8 , 9 , I 1
(*)(IS) 1. 4, 5. 13, 12, 1 1 , 14, 15. 8
(*)(16) 1, 8, 9, 11, 10, 2, 13, 12. 3
(*)(18) I , 8 , 9 . 11. 10, 2, 13, 12. 3
(20) 1, 6, 7, 12, 13, 10, 14, 15, 1 1
(21) I , 2, 3. 5 , 4, 6, 13, 12, 9
(22) 1, 2. 3. 5 , 4 , 6 , 11. 1 0 , 8
(23) 1, 2, 3. 4, 5 . 7. 14. 15, 13
(24) I , 2, 3, 4, 5 , 7, 13. 12, 14
(25) 1 , 2, 3, 4, 5. 7, 13, 12. 14
(26) I , 2, 3 , 4 , 5, 7, 10, 11. 9
(*)(27) 2, 8, 10, 1 1 , 9, 1, 15, 13, 6
(*)(28) 1, 8, 9, 11, 10, 2, 14, 15, 6
(29) 1 , 6 , 7 , 3 , 2 . 4 , 9 . 8 , 1 4
(30) 1, 6, 7, 3, 2, 4, 11, 10, 14
Projectiue embeddings of small Steiner Triple Systems
157
(31) 1, 6, 7, 3, 2, 4, 11, 10, 14
(*)(32) 1, 6, 7, 3, 2, 4, 14, 15, 10
(33) 1, 6, 7, 3, 2, 4, 14, 15, 8
(*)(34) 1, 6, 7, 3, 2, 4, 11, 10, 14
(35) 1, 4, 5 , 9, 8, 3, 10, 11, 13
(*)(38) 1, 6, 7, 5 , 4, 2, 15, 14, 9 (40) 1, 4, 5, 7, 6, 2, 10, 11, 14
(*)(39) 1, 4, 5, 7, 6, 2, 9, 8, 3
(42) 1, 12, 13, 8, 9, 7, 15, 14, 2
(41) 1, 4, 5, 7, 6, 2, 9, 8, 3 (*)(45) 2, 11, 9, 8, 10, 1, 12, 14, 7
(*)(46) 1, 4, 5, 7, 6, 2, 9, 8, 3
(*)(50) 2, 5, 7, 10, 8, 14, 11, 9, 12
(*)(48) 1, 6, 7, 8, 9, 14, 4, 5 , 2 (*)(51) 1, 12, 13, 15, 14, 2, 8, 9, 7
(*)(52) 1, 10, 11, 4, 5, 12, 9, 8, 2
(*)(53) 2, 5, 7, 6, 4, 1, 13, 15, 8
(*)(54) 1, 4, 5, 7, 6, 2, 8, 9, 15
(*)(55) 1, 8, 9, 6, 7, 14, 11, 10, 2
(*)(56) 1, 6, 7, 5, 4, 2, 12, 13, 8
(*)(58) 4, 7, 12, 15, 10, 9, 11, 14, 8
(*)(59) 1, 4, 5 , 7, 6, 2, 8, 9, 12
(*)(60) 2, 9, 11, 10, 8, 1, 13, 15, 4
(*)(65) 2, 5, 7, 6, 4, 1, 13, 15, 8
(*)(67) 1, 4, 5, 7, 6, 2, 12, 13, 14
(*)(68) 1, 4, 5, 7, 6, 2, 9, 8, 3
(*)(70) 2, 5, 7, 8, 10, 12, 11, 14, 3
(47) 1, 4, 5, 7, 6, 2, 9, 8, 3
The above list, also established with the aid of a computer, takes care of 16 new systems. (The mark (*) signifies that the system had already been eliminated in 3.1 or 3.2)
3.4. We have treated the 14 remaining S(2,3, 15) in the same way as the two S(2,3, 13). We will write B(a, b, c) to mean that {a, b, c} is a block in the system we consider. System No. 7. This system contains three subsystems S(2,3,7) (see Fig. 4). Therefore if it is embeddable into a projective plane P, P must have characteristic
Fig. 4.
M. Limbos
158
2. We may assume that: l=(l,O.O), S = ( l , O . l ) ,
9=(a+A,p,l),
13=(y+p,6,1),
2 = ( 0 , 1.0). 6 = ( 0 , 1 , l ) ,
10=(a,p+A, l),
14=(y, 6 + p , l ) ,
3=(1, I,()), 4=(0,0, l ) ,
7 = ( 1 , 1 , I ) , I l = ( a + A , p + A , I ) , 1 5 ~ ( y + p , f j + p1 ,) X = ( a , P , 11, 12=(y, 6, l ) ,
where a, p, y, 6, A,
p
are non-zero parameters. Then
B(4,8, 1 2 ) J a S = py, (1)
B ( 5 , 10. 12) and ( l ) + p + A + f j + y A = O , (2) B ( 6 , 1 1 , 12) and ( l ) J a + A + y + t i A + y A = O ,
(3)
R ( 7 . 1 1 . IS) and ( l ) J a + p + y + f j + p a + p P + A 6 + A y = O ,
(4)
B(4. 10. IS) and ( l ) j a p + P p - A y + A p = O . ( 5 ) Taking the sum (2)+ (3)+ (4)+ ( 5 ) we get Ap and # 0.
= 0,
which is impossible since A # 0
Syslcrn No. I Y . I t contains a subsystem S ( 2 . 3, 7 ) , and so the characteristic of P is 2. We may assume that (see Fig. 5 ) : I = ( l , O , O ) . 4=(0.0.1).
7 = ( 1 , l , I),
2 = ( 0 . I,()),
SE(1.0, I),
8 = ( a , p, l ) ,
3 = ( 1 . l.O),
6 = ( 0 . 1. 1 ) .
9 = ( a + u a , p ,1 )
9
4
1
2
3
Projecfive embeddings of small Sfeiner Triple Systems
159
B(4, 11, 13) and B(5,8, 1 3 ) j 1 3 = ( ( a + a ) ( l + a + a ) ,@ ( l + a ) ,1 + a + a + a 2 ) , B(3, 10, 13) and ( l ) J l + a + a ’ = O ,
(2)
B(1, 12, 1 3 ) j P = a + a + a a . (3) Multiplying (3) by 1 + a which is different from 0 (otherwise a would be 0) and using (2) we get aa = 1. Thus (3) gives 1+ a + a = 6, which contradicts (1) since a#O.
System No. 36. Thanks to the configuration in Fig. 6, we may assume that
1=(O, 0, l),
4=(1,0, l), 7 = ( 0 , 1, l),
2=(1,1, l),
5=(1,0,0), 9=(p, l , a ) , p # 1,
3 ~ ( 1l , a ) , a f l , 6 ~ ( 0l ,,O ) ,
Sz(p,l,p+a-l).
Fig. 6 .
Observe that a # 0 (because {3,5,6} is not a block) and p # 0 (because 9 is not collinear with 1, 6, 7).
B(5,8, 13) and B(3,7, 1 3 ) j 1 3 = ( / 3 + a - 2 , a - 1 , ( a - l ) ( @ + a - - l ) ) , B(1, 12, 13) and B ( 3 , 6 , 1 2 ) 3 1 2 E ( p + a - 2 , a - 1 , a ( P + a - 2 ) ) , B(4,9,12)=,2(a-l)(l-p)=0. Thus the characteristic of P is 2.
B(6,9, 10) and B(4, 10, 13)+ lO=(pa, a + 1, a’), B(1, 10, 11) and B(6, 11, 1 3 ) $ 1 l = ( p a , a + l , a p ( l + a ) ) , B(5,11, 12)Ja2=0. This is clearly a contradiction since a # 0.
M. Limbos
160
Sysrem No. 37. This system contains exactly the same configuration as the one exhibited above in system No. 36. B(3,6, 12) and B(7,9, 12)+ 12=(@,ap - a B ( l , 12,13) and B(6,8,13)
+ 1, a @ ) ,
+ 13=((p, a @ - a + 1,@ + a - l),
8(3,7, 1 3 ) + 2 ( a - l ) ( P - l ) = O . Thus the characteristic of P is 2.
B(5,10,12) and B(4,7,lO)jlO=(a+l,ap+a+l,ap), B(5,6, 11) a n d B ( l , l O , l l ) j l l ~ ( a + l , a @ + a + l , O ) ) 0. B(2,9, 1I ) + ( 1+ a + p)(1+ a @ =
If 1 + a + p = 0, the four points 5, 6, 11, 13 would be collinear which is impossible, and so a p = 1. But B(8, 11, 1 2 ) j p + a = O , that is p = a . Therefore a’= 1. Since the characteristic of P is 2, this implies a = 1, a contradiction.
System No. 43. This system contains the configuration, shown in Fig. 7. The coordinates of these points are given by a simple change of notation in system No. 36. B(9, 10, 12) and B(4, 11, 12)+12=(1-a,p,O), B(1. 12, 13) and B(3, 11, 1 3 ) j 1 3 = ( 1 - a , p , ( l - a ) ( l - p ) ) , B(5,8, 13)j ( a - 1 + P)(a- 2) = 0. But a + p - I f 0 (otherwise 5, 8, 9 and 12 would be collinear), and a - 2 2 0 (otherwise the coordinates of 10 would be (p, 1, p + 1) and {8, 5, 10) would be a block which is not the case). This is the required contradiction.
Fig. 7.
Projective embeddings of small Steiner Triple Systems
161
System No. 44. This system contains exactly the same configuration as the one exhibited in system No. 43.
B(6,9,10)andB(2,4,6)+6=(a-l,a-P,O), B(1,6,7) and B(7,8, l l ) + 7 = (a- 1,a - p, 2a - p - l), B(2,5,7)+2(0-1)(/3-1)=0. Thus the characteristic of P is 2. B(4,9, 12) and B ( 5 , 1 1 , 1 2 ) j 1 2 ~ ( a + p , p ( a + l ) , a ( a + p ) ) , B(3,6, 1 2 ) j a 2 ( a+ 1)(p+ 1)= 0. This is a contradiction since a # 0, a # 1 and p # 1. System No. 61. This system contains exactly the same configuration as the one exhibited in system No. 19. Therefore the characteristic of P is 2.
B(3,8,11) and B ( 4 , 9 , 1 l ) j 1=(ay, p, y), (so that y f 1) B(1, 10, 11) and B ( 2 , 8 , 1 0 ) 3 l O ~ ( a y , p , P + y + P y ) , B 6 9 , 1 0 ) j a y = p2, (1) B ( 4 , 8 , 12) and B(2,9,12)+12=(a(l+p), y ( a + P ) , y+p), (using (1)) ~ ( 5 , i i1, 2 ) j p 3 + ~ 2 ( i + y ) + y 2 + p y = (2) 0, ~ ( 7 , i o1,2 ) j p 3 + p 2 ( 1 + y ) + y 2 + p y ~ = o . (3) Taking the sum (2)+ (3) we get Py( 1+ y ) = 0. This is a contradiction since /3 # 0, y # o and y f 1. Systems No. 62 and 63. Both systems contain the configuration in Fig. 8. Note that the characteristic of P is different from 2 otherwise {3,5,6}would be a block. We may assume that: 1=(1,0,0),
6=(0, 1, l),
2=(0, 1,0),
7=(1, 1, l),
3=(1,1,0),
12=(1,ar,a+l) with a Z 0 ,
4=(0,0, l),
l l = ( l , P + l , p ) with p#O,
5 ~ ( 1 , 0l), ,
S=((Y-~,(Y,-~).
B(1, 10, 11) and B(2,8,10)+10=(p--a,P+1,/3), B(7,10, 1 2 ) 3 2 a =O+a =0, since the characteristic is not 2. This is the required contradiction.
M. Limbos
162 1
12
Fig. 8 .
System No. 64. This system contains a configuration similar to the one described in systems No. 62 and 63, except that the points 1 1 and 12 above are now 13 and 1 1 respectively.
B ( 1 , 8 , 9 ) and B(4,9, 13)5$9=(a, a ( @+ 11, - p ( p
+ l)),
B(7,9, l l ) j ( ~ ’ p = ( l - ~ ~ ) ( c+pZ), ~ + p (1) B(1.12, 13) and B ( 4 , 1 1 , 1 2 ) + 1 2 = ( / 3 + 1 , a ( p + l ) , a p ) , B(7,8,12)Ja’P = ( l - a ) ( a + p Z ) + ( l + a ) p .
(2)
( 1 ) and (2) give I + a = 1 - a , that is 2a = 0 which is a contradiction since the characteristic of P is not 2. System No. 74. This system contains the configuration of Fig. 9, and so we may 1
6
Fig. 9.
Projecrioe embeddings of small Sreiner Triple Sysrems
163
assume that:
1=(1,0,0), 4=(1, l , O ) ,
8 = ( a , p , l),
2=(1, 1, l), 6=(0,0, l),
9 = ( a + y , P , l),
5=(0, l,O), 7 ~ ( 1 , 0l), , l l = ( a , p , 1-y) where a, p, y are non-zero parameters. Observe moreover that y # 1, p # 1, a f l and a + y # O B(5,6, 13) and B(4, 11, 13)+13=(0,p-a, 1-y), B(1,2,3) and B ( 3 , 4 , 8 ) + 3 = ( a - p + l ,
1, l),
B(5,8, 15) and B(6,9, 1 5 ) j 1 5 ~ ( a ( a + y ) , c u S , a + y ) , B(4,7, l S ) + ( a - l ) ( a + y ) = a p ,
(1) (so that 1 5 = ( a , a - l , 1))
B(1, 10, 11) and B(5,9, lO)+lO=((a+y)(l-y),p, 1-y), B(7, 1 0 , 1 3 ) + ( l - y ) ( a p - ( ~ * - ~ ~ ~ + ( ~ + p y ) =(2) O. Since y f 1, (1) and (2) give By = y, a contradiction because p # 1 and y f 0. System No. 76. This system contains a Desargues’ configuration on the points 1, 2, 4, 5, 7, 9, 10, 11 and 12, so that we may assume that: l = ( l ,O,O),
7=(1, 1, 0),
2=(1, 1, l),
9 = ( a , P , 1-y)
with a#O,p#O, y f 0 ,
4=(1,0, l), lO=(a+y,P, I),
a+y#O,
5 = (0, 0, l), 11 =(a,p, l),
P+yfO,
6=(0, 1,0), 1 2 = ( a + y , P + y , 1).
Y f 1
B(5,6, 13) and B(1, 12, 13)$13=(0, p + y , I), B(1,2,3) and B(3,6, 1 1 ) 3 3 = ( a , 1, l), (so that a # 1) B(3,7,9)+ 1 - y - p
=O,
(1)
B(4, 11, 1 3 ) + - 7 + ~ ~ + a P = O ,(2) B(3, 10, 13) and ( l ) j ( a + y - l ) a y = O .
(3)
Therefore 1y = 1- y (by (1) and (3)) and {2,9,6} must be a block which is a contradiction. System No. 79. This system contains the configuration of Fig. 0 and so we may assume that:
l=(l,O,O),
5 ~ ( 1l ,,O ) ,
8 3 ( ~ , p1-y), ,
2=(1,1, l), 6=(1,0, l), 12=(a,p, l), 4=(0, l , O ) ,
7=(0,0, l), 1 3 = ( a + y , p , 1)
where a,p, y, a + y and 1 - y are all different from 0.
M. Limbos
164
1
6
Fig. 10
8 ( 1 . 2 , 3 ) and B ( 3 . 4 , 8 ) 3 3 1 ( a , l - y , l - y ) , B(4, 7, 11) and B ( 6 , 11, 1 2 ) j 11 = (0,p, 1 - a ) ,
(so that y # l ) (so that a # 1)
8 ( 3 , 7 , 1 3 ) 3 a P = ( l - y ) ( a + y ) , (1)
B(5. 11, 1 3 ) j a p = ( ~ ~ - l ) ( a + y )(2) . Taking the difference ( 2 )- (1) we get ( - 2 + Q + y ) ( a + y ) = 0 which is equivalent to 2=a+y. B( 1.8.9) and B(5,6,9) j9 = ( p + a - 1 , p, a - I ) , B(3,9,1 l ) + a * +p’- 2a + 1 - 2 p
= 0,
(3)
B(2.9. 12)j a*+ p2 - 2a + 1 - ( ~ =p0. (4) (3) and (4) imply a = 2, so that y = 0, a contradiction. System
No. 80. This system contains the configuration of Fig. 11, so that we may
1
3
7
Fig. 1 1 .
Projective embeddings of small Steiner Triple Systems
165
assume that: l=(l,O,O), 4=(1,0,1), 6 = ( a + l , a , l ) , 2=(1, l,O), 5=(0,0, l), 7=(0,a, 1) with a#O and a#-1. 3 = ( 0 , 1, O ) ,
B(2, 5 , 8 ) 3 8 = ( 1 , 1, p ) with @ # O , B(1,8,9) and B(2,7,9)+9=(1-ap,
l,p),
B(3,4, 10) and B(5,9, l O ) j l O = ( l - a @ ,
1, 1-ap),
B ( 5 , 6 , 14) and p(4,9, 1 4 ) + 1 4 = ( a + l , a , l + a p + a Z @ ) , B(8, 10, 1 4 ) j 1 = P 2 ( l - a 3 ) , (1) B(1, 10, 11) and B ( 3 , 6 , l l ) + l l = ( ( a + l ) ( l - a p ) , 1, 1-ap), B(7,8, 11)+ (2 - aP)(1- p - ( ~ p=)0. (2) If 1 - p - a@ = 0, the points 9 and 10 would not be distinct, and so ap = 2
B(2, 11, 1 4 ) 3 2 a = - 5 ,
and
(by (2)) - 5 p = 4 .
Using (1) we find that the characteristic of the projective plane P must be 241. B(1, 14, 15) and B ( 3 , 8 , 1 5 ) j 154-55,20,44), B(4, 11,15)+298=0, a contradiction since the characteristic of P is 241.
0
In this note we have been concerned with the embeddability of an S(2,3, u ) in a finite Desarguesian plane P. Note that our result remains true if P is supposed to be Pappian (not neccessarily finite). If P is Desarguesian, but not Pappian, the 56 systems which contain a configuration contradicting Desargues' theorem are certainly not embeddable in P. For the remaining 24 systems the problem is open. If P in non Desarguesian . . . the problem is open too!
References [ l ] P. Dembowski, Finite Geometries (Springer, Berlin, 1968). [2] J. A. Thas, Connection between the n-dimensional affine space A,,,, and the curve C, with equation y = x4, of the f i n e plane A2.qn,Rend. 1st. di Matem. Univ. di Trieste. Vol. I1 fasc. I1 (1970).
[3] J. Hirschfeld, Projective geometries over finite field, Chap. 5 (Oxford University Press, 1979). 141 F. C. Bussemaker and J. J. Seidel, Symmetric Hadamard matrices of order 36, Technological University Eindhoven Report 70 WSK-02 (1970).
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167
Projecfiue embeddings of small Steiner Triple Systems Appendix m. 12
N. 1 1
1 1 1
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3 3 3 3 3 3 L
b
L 4 5 5 5 5 6
6
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M.Limbos Appendix m . 21 1 l 1
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m. 29
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28
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llr
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8
10 11 11
Projective embeddings of small Steiner Triple Sysfems
169
Appendix N.
1 1 1
1
8
1 1 1 2 2 2 2 2 2 3 3
10 12 14 4 5
3
6
3
9 10 11
3 3
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m. 311
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M . Limbos
170
Appendix m-.
2
3
1
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m. 44
m. 43
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10
13
1
8
1
7
10 11
13 lh 12
8
12
9
10
15 15
171
Projective embeddings of small Steiner Triple Systems
Appendix m. 52
51
N.
2 b 6 8
1 1
10
11
1
12 1L h 5
13 15 6 7 10
1 1
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Annals of Discrete Mathematics 7 (1980) 175-202 @ North-Holland Publishing Company
A SURVEY OF EMBEDDING THEOREMS FOR STEINER SYSTEMS* C.C. LINDNER Mathematics Deportment, Auburn University, Auburn, AL 36830, U S A
1. Intrcxluction A S(t, k, u ) Steiner system is a pair (S, B) where S is a set containing u elements (points) and B is a collection of k-element subsets of S (blocks) such that every t-element subset of S is contained in exactly one block of B. The number IS1 = u is called the order of the Steiner system (S, B). A S(2,3, u ) Steiner system is called a triple system, any S(2, k, u ) Steiner system (including k = 3) is called a block design, and a S(3,4, u ) Steiner system is called a quadruple system. In this paper we will concern ourselves solely with triple systems, block designs, and quadruple systems. A partial S ( t , k, u ) Steiner system is a pair (P, p) where P is a set of u points and p is a collection of blocks of size k such that every t-element subset of P is contained in at most one block of p. Given a partial S ( t , k, u ) Steiner system (P,p) a very natural thing to attempt is to try to add blocks to the collection p so that the resulting collection p* has the property that (P, p*) is a S ( t , k, u ) Steiner system. In general this cannot be done. For example, the pair (Q, q) given by:
I
Q ={I, 2,3,4,5,61, q
= {{1,2,31, {4,5,611
is a partial triple system. However, the collection of triples q cannot be enlarged t o a collection q* so that (Q, q*) is a triple system since (among other reasons) IQJ = 6 z& 1 or 3 (mod 6 ) , which is a necessary condition for the existence of a triple system. Since a partial Steiner system cannot always be completed to a Steiner system the following question is of considerable interest: given a partial S ( t , k, u ) Steiner system (P, p), is it always possible to find a S ( t , k , w ) Steiner system (Q, q) such that P E Q and p 5 q? That is to say, is it always possible to embed a partial S ( t , k, u ) Steiner system (P, p) in a (hopefully not too large) S ( t , k, w ) Steiner system (Q, q)? In 1971, in a by now classic paper, Christine Treash [32] proved that any partial triple system can always be embedded in a triple system. Treash’s Theorem that a partial triple system can always be embedded in a triple system
* Research supported by NSF Grant
MCS 77-03464 AOl. 175
176
C.C. Lindner
was the starting point for a large collection of results on various kinds of embeddings of (partial and complete) triple systems, block designs, and quadruple systems. The object of this paper is t o give a (hopefully) fairly comprehensive survey of a large subset of the present day state of the art. The author frankly admits that this paper is uneven; i.e., some topics ate covered in more detail (much more detail in some instances) than others and some topics are omitted. This, of course. has t o d o with the author’s interests. And anyway, to use CATCH-22 terminology, it’s always never possible to cover everything (evenly or otherwise).
2. Treash’s Theorem Since Christine Treash’s result that a partial triple system can always be embedded in a triple system is the genesis of the study of embedding theorems for block designs, it is certainly the place to begin a survey on the subject. Although Treash’s Theorem has subsequently been improved upon with respect to the size of the containing triple system (see Section 5 ) we give here Treash’s original proof. There are two reasons for this: The first is the historical importance of the result, and the second is the motivation for the techniques used later by B. Ganter [5,6,7] and R. W. Quackenbush [28] to embed partial block designs with block sizes larger than 3 (these results have not been improved upon). It is well-known that there is a triple system of order n if and only if n = 1 or 3 (mod 6 ) [16] and that if (S, t ) is a triple system of order n that It1 = i n ( n - 1). Triple systems have the so-called “replacement property”. That is, if (S, t ) is any triple system containing the subsystem ( P , pl). and (P, pz) is any triple system, then (S, ( t \ p l ) U p , ) is a triple system. The transition from ( S , t ) to (S, (t\pI)Up,) is usually referred to as unplugging the subsystem ( P , p,) and plugging in (P, p2). Finally, almost everyone’s favorite triple system is the projective plane of order 2 given by S = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } and t={{1,2,4}, {2,3,5}, {3,4,6}, {4,5,7}, {5,6, l}, {6,7,2}, (7, 1,3}}. (We remark that the projective plane of order 2 is the unique triple system of order 7.) Treash’s proof is based o n the following well-known construction for triple systems (the so-called u --+ 2u + 1 construction): Let (S, t ) be any triple system, 03 an element not in {2,3}x S, K = {CQ}U({2,3} x S), and define a collection B of triples of K as follows: ( 1 ) The IS1 triples {CQ,(2, x), (3, X)}E B for every x E S; and (2) For every triple {x, y, Z } E t define a copy of the triple system of order 7 ( = projective plane of order 2) on the set {m} U ({2,3} x {x, y, z}) with the proviso that the triples (00, (2, XI, (3, x)}. {CQ, (2, Y), (3, Y)},{m, (2, z ) , (3, z)}, ((2. XI,(2, Y), ( 2 , z)} belong to the collection, and place these triples in B. It is a routine matter so see that ( K , B ) is a triple system which contains an isomorphic copy of (S,t ) , since {x, y, Z } E t if and only if ((2, x), (2, y), (2, Z ) } E B.
Embedding theorems for Steiner systems
177
Important: In what follows to keep the notation from getting out of hand we will abbreviate (2, x ) to x so that, with this convention, (S, t ) is literally embedded in ( K , B). Finally, we will refer to the triple system of order 7 defined in (2) above as the subsystem of (K, B) generated by {x, y , z}.
Lemma 2.1. Let ( S , t ) be any triple system and b = { x , y , z } any triple in t. Then ( S , t\{b}) can be embedded in a triple system ( K ,B ) containing a triple { x , y , z*} where z*EK\S. Proof. Embed (S, t ) in a triple system using the u + 2 u + 1 construction given above. Let (P, p ) be the subsystem of order 7 generated by the triple { x , y , z } . Then (P, p ) contains, of course, the triple { x , y , z } but no triple belonging to t\{b}. Hence if we unplug (P, p) and replace it with the triple system (P, p * ) obtained by interchanging m and z in the triples of p , the resulting triple system (K, B) not only contains the triples in t\{b}, but also a triple of the form { x , y , m} where m€K\S.
Theorem 2.2. (C. Treash [32& A partial triple system can always be embedded in a triple system. The proof is by induction on the order of the partial triple system. Since a triple system of order 1can trivially be embedded, we can assume that any triple system on less than u points can be embedded and that (P, p ) is a triple system of order u. Let z E P and denote by b, ={xl, y l , z } , b, = z}, . . . , b k = { x k , Y k , z} the set of all triples in p containing 2. By induction (P\{z}, p\{b,, bZ,. . . , bk}) can be embedded in a triple system (S, t). Let b, = {xl, y l , ZJ, . . . , b k = { x k , y k , z k } be the triples in t containing the pairs { x l , y l } , {x,, y,}, . . . , { x k , Y k } . By repeating Lemma 2.1 k times we can embed (s,t\{b,, b,, . . . ,b k } ) , and therefore (P\{z}, p\{b,, bz, . . . ,bk}),in a triple system ( S * , t * ) containing k triples of the form by ={xl, y l , z;}, b; = * } ,. . . ,b: = {xk, y k , z}: where the 2:’s are distinct and none belong to S {x,, y,, z 2 (and therefore none belong to P ) . Now embed (S*, t * ) in a triple system using the u + 2 u + 1 construction and denote by (Qi,qi) the subsystem generated by bT = { x i , yi, 2:). Now, the k subsystems (Qi, q i ) have no triple in common (since bT n by = $3, for all i, j ) , intersect pairwise in the point w, and contain no triple belonging to t*\{by, b;, . . . ,bc}. Unplug each (Q, qi) and replace it with the triple system (Q, qT) obtained by interchanging m and zT in the triples of 4,. Call the resulting triple system ( K , €3). Then not only does B contain the triples in p\{b,, b2, . . . ,bk} since p\{b,, bZ,. . . , b k } st*\{bt, b f , . . . , bf)) but also k triples of the form {q,y , , m}, {x,, y 2 , m}, . . . ,{ x k , Y k , m} where, of course, m $ p. Hence renaming m with z imbeds (P, p) in the resulting triple system.
Proof. partial partial partial {x,, y,,
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3. Embeddmg block designs With Treash's Theorem in hand, the problem of embedding partial block designs with block size k 8 4 is immediate. Again, to begin, it is well-known that a necessary condition for the existence of an S ( 2 , k, u ) Steiner system is u - 1= 0 (mod ( k - 1)) and u ( u - 1)= 0 (mod k ( k - 1)). Although these conditions are not in general sufficient, they are sufficient for the existence of an S ( 2 , k, u ) block design for k = 3 (Steiner triple systems (T.P. Kirkman [16])), for k = 4 and 5 (H. Hanani [ l l , 121). and for k a 6 for sufficiently large u (R.M. Wilson [341). An obvious first attempt at settling the embedding problem for partial block designs with block size k 8 4 is the use of Treash's technique for embedding triple systems. Unfortunately, this technique fails (or, at least, the author does not see how to make it work short of bedlam) for the following reason: if (P,p) is a partial block design with block size k 24 and b l , b,, . . . , bk are all blocks of p containing the point z, then using induction t o embed (P\{z}, p\{bl, b,, . . . , bk})in a block design ( S , 1 ) does not guarantee that each of the sets bi \ { z } is a subset of a block of t (it does, of course, for triple systems since Ib,\{z}l=2). In [ S ] Bernhard Ganter got around this problem for k = p " + l ( p a prime) by changing the induction assumption from the number of points to the number of blocks. Before proceeding any further we will need the following (again well-known) generalization of the u + 2 u + 1 construction for triple systems (referred to hereafter as the 2' -+( k - 1) u + 1 construction). Let k and r be positive integers such that there exists a S ( 2 , k , ( k - l ) r + l ) Steiner system and let (S, T ) be any S ( 2 . t . u ) Steiner system. Let be an element not in {2,3,. . . . k} x S, K = {w} U ( { 2 , 3 ,. . . , k } X S ) , and define a collection B of blocks of K as follows: (1) The IS1 blocks {m, (2, x), (3, x), . . . ,( k , X)}E B for every x E S; and (2) For every block {x,, x 2 . . . . , X , } E T define any S ( 2 , k, (k - 1)t + 1) Steiner system on {m}U({2, 3 , . . . , k } x { x , , x,, . . . ,x,}) with the proviso that the blocks {=, (2, x,), (3. x ~ ) ., . . , ( k , x i ) } ; i = 1,2,. . . , t ; belong to the collection, and place these blocks in B. It is immediate that ( K , B) is a block design with block size k. In what follows we will refer to the Steiner system defined in (2) as the subsystem of ( K , B) generated by the block {x,, x 2 , . . . , a} of T. As was the case with triple systems we will abbreviate (2, x ) to x for all x E S. If f = k, then the subsystem generated by {xl, x2. . . . , x t } is, of course, a projective plane of order k - 1. In this case we will always require that {x,, x 2 , . . . ,X k } is a block of this subsystem. So in the case where t = k the u + (k - l ) u + 1 construction embeds (S, T) in the resulting block design (K. B). Finally, as was the case with triple systems, block designs have the replacement property. That is, if (S, t ) is any block design containing the subsystem ( P , p , ) and (P, p,) is any block design (with the same block size) then ( S , ( t \ p,) U p2) is a block design.
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Lemma 3.1. Let k = p a + 1 ( p a prime) and let ( S , t ) be any S ( 2 , k, u ) block design. If b is any block in t and X is any subset of b containing at most k - 2 points, then ( S , t \{b}) can always be embedded in a block design containing a block of the form ( b\ X ) U X * , where none of the points in X* belong to S. Proof. It suffices to show that any block design containing (S, t\{b}) and a block of the form ( b\ X) U X, where )XIn Sl = j can be embedded in a block design containing (S, t\{b}) and a block of the form ( b \ X ) U X 2 where IX,nSl<j. So let ( K ,B) be a block design containing (S, t\{b}) and a block of the form b, = (b\X) U X, where IX, fl SI = j. Let S(2, k, ( k - l ) k + 1) be any projective plane and embed ( K , B) in a block design using the u -+ (k - 1)u + 1 construction given above. Let x E X , f l S and denote by ( P , p ) the subsystem (=projective plane of order k - 1) generated by the block b,. Then (P, p ) contains the block b, but no block belonging t o B \{b,} and therefore no block belonging to t \{b}. Hence if we unplug (P, p ) and replace it with the block design (P, p*) obtained by interchanging x and 03 in the blocks of p, the resulting block design still contains the blocks in t \{b} and also a block of the form b2 = ( b \X) U ( X I\{x}) U(00). Taking X 2 = (XI\{x}) u (00) gives Jx,n SI <j .
Theorem 3.2. (B. Ganter [5]). A partial block design with block size k = p a
+1
(where p is a prime) can be embedded in a block design.
Proof. Let (P, p ) be a partial block design with block size k = p a + 1 ( p a prime). The proof is by induction on the number of blocks Ip). If (p(= 1 the result is trivial and so we can assume that (P, p ) can be embedded wherever 1 S Ip) < n and that 1p1= n. So let b be any block in p and (S, t ) any block design containing ( P , p \{b}). If b E t we are through. Otherwise, let c be any block in t such that I b n c l = j is maximal. We will now embed (S, r) in a block design containing ( P , p\{b}) and a block of the form c’ where Ib nc’l= j + 1. Iteration of this embedding eventually gives the desired embedding. So write c = ( b n c ) U ( c\ b ) , let x E b\c, and denote by b2, b3,. . . ,b, the blocks of p other than b which contain x. By repeated applications of Lemma 3.1 we can embed (S, t\{c, b2,. . . , b,}) and therefore (P, p\{b, bZ,b3.. . . , b,}) in a block design containing pairwise disjoint blocks of the form bT = ( b n c ) UX,, b: = (bz\{xN U{x21, . . . , b: = (b, \{XI> U{x,,,l where the sets X , , b21, { x d , . . . ,{x,,,} contain no points of S, and therefore none of P. Now use a S(2, k, ( k - l ) k + 1) projective plane to embed this block design into a block design ( K , B) using the u + ( k - l ) u + l construction and denote by (Qi,qi)the subsystem of ( K , B ) generated by the block bT. Then: the subsystems (Q,,41),(Q2, q2).. . . ,(om, 4,) have no block in common, intersect pairwise in the point 03, and contain no blocks belonging to (P, p\{b, b2,. . . , b,}). Now unplug each (Q,,q ) and replace it with the block design (Q,,47) obtained by interchanging 00 and xi in the blocks of Q. where x , is any point in X I . Call the resulting block design ( K , B * ) .Then B*
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contains the blocks in p\{b, b,, . . . ,b,} and also m blocks of the form (b n c ) U ( X l \ { x l ) ) U { ~ } (, b , \ { x } ) U { ~ } ,. . . , ( b , \ { x } ) U { ~ } . If we now rename 00 with x, then the resulting block design not only contains (P,p\{b}) but also the block c' = (b n c ) U ( X ,\{x,}) U { x } . Trivially Ic'n b( = j + 1.
'lhorern 3.3. (R.W. Quackenbush [28J). A partial block design wirh block size p a (where p is a prime) can be embedded in a block design.
Proof. Let ( P , p) be a partial block design with block size k = p a (p a prime). Denote the blocks in p by b , , b,. . . . , b,, and let X = { x ] , x2,. . . , x,,} be a set of n distinct points none of which belong t o P. Now if we set bT= b , U { x , } , b ; = b z U { x , } , . . . , b ~ = b , , U { x , , }and define p * = { b : , b ; , . . . , b i } , then ( P U X , p * ) is a partial block design with block size k + 1 = p " + l . Hence by Theorem 3.2 ( P U X , p*) can be embedded in a S(2, k + 1, u ) block design (S, T). Now using a S ( 2 , k, ( k - l ) ( k + 1)+ 1) affine plane construct an S ( 2 , k, ( k - l ) v + 1) block design ( K , B) from ( S , T) via the u + (k- l ) u + 1 construction. If in constructing ( K , B) we make sure that the block bi belongs to the subsystem generated by bT (this is trivially possible), then the original partial block design ( P , p) is embedded in ( K ,B ) . Before proceeding to the main result in this section we will need the following observation. First if k is a power of a prime, then the necessary condition for the existence of a S ( 2 , k , v ) block design can be combined to u s 1 or k (mod k ( k - 1)). Since the arithmetic progression { k ( k - 1)t + 1 1 t = 1,2,3, . . .} contains infinitely many primes, and the conditions u = 1 or k (mod k ( k - 1)) are asymptotically sufficient for the existence of a S(2, k, v ) block design [34], it follows that for every positive integer k there is a prime p > k such that a S ( 2 , k, p ) block design exists. This last remark is crucial to the result which now follows.
Tbeorem 3.4. (B. Ganter [7]). A partial block design can always be embedded in a block design.
Proof. Let ( P , B) be a partial block design with block size k and let p > k be any prime such that a S(2, k, p) block design exists (see remarks preceding the statement of the theorem). Denote the blocks of B by b l , b 2 , . . . , b, and let X , , X , , . . . , X,, be n pairwise disjoint sets each of which is disjoint from P and such that / X , I = l X z l = . . . = I X , , I = p - k . If we set X = X , U X , U . . . U X , , b y = b, U X , , and B* ={bT,b:, . . . ,b:}. then ( P U X , B*) is a partial block design with block size p (a prime). Hence by Theorem 3.3 ( P U X , B*) can be embedded in a block design ( K , T ) (with block size p, of course). Now defining a S(2, k, p) block design on each block of T, with the proviso that the S(2, k, p) design defined on each of the blocks b y contains the block b,, embeds ( P , B ) in ( K , T).
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Remarks. Actually Ganter proved a more general theorem than the one given here (Theorem 3.4). In [7] Ganter proved that any partial painvise balanced design (PBD) with block sizes from a set K , can always be embedded in a PBD whose block sizes also come from the set K. However, we are interested here in block designs only and so refer the reader to [7] for details of the proof of this more general result. Finally, we remark that a different proof of Ganter’s Theorem 3.4 due to R. M. Wilson can be found in [33]. Here Wilson proves that for a given graph G (undirected with no loops or multiple edges, of course) the trivial necessary conditions for the complete graph K,, to be decomposed into edge disjoint isomorphic copies of G are asymptotically sufficient. Since a (partial) S(2, k, u ) block design is equivalent to an edge disjoint decomposition of (a subgraph of) & into copies of Kk,Wilson’s Theorem can be used to prove Ganter’s Theorem 3.4 as follows: Let (P, B) be a partial block design with block size k and let G be the graph with vertex set P and edges [x, y] if and only if {x, y} belongs to a block of B. Now decompose any complete graph K,, into copies of G such that one of the copies of G is the graph corresponding to the blocks of B and then decompose each copy of G into the copies of Kk defined by the blocks of B. Trivially this procedure embeds (P, B) into the resulting S(2, k, n) block design. A detailed proof of Wilson’s graph decomposition theorem includingthe background is not really any shorter than the proof given here, which is one reason for presenting Ganter’s proof. However, the main reason for using Ganter’s proof is that the background and techniques used by Ganter (and Quackenbush) are used repeatedly in subsequent parts of these notes.
4. Embedding quadruple systems In 1960 H. Hanani [lo] proved that the spectrum for quadruple systems is the set of all u = 2 or 4 (mod 6). In 1974 B. Ganter [6] proved that a partial quadruple system can always be embedded in a quadruple system. We give here a slightly different proof of Ganter’s Theorem. The following well-known doubling construction for quadruple systems is the main tool used in the proof. Let (X, x) and (Y, y) be any two quadruple systems of order u where Xfl Y = Let$ F={Fl, IF,, . . . ,Fu-l}and G ={GI, G,, . . . ,GvPl} be any two 1-factorizations of K,, (the complete v-graph) based on X and Y respectively, and let a be any permutation on the set {1,2, . . , ,u - 1).Define a collection of blocks q on Q = X U Y as follows: (1) Any block belonging to x or y belongs to q, and (2) If x , , x , ~ X a n d y 1 , y 2 ~ Ythen(x,,x,,y,,y,}Eq , ifandonlyif [ x , , x , ] E ~ ~ , [yl, y,] E Gi, and ia = j. It is a routine matter so see that (Q, q) is a quadruple system. (We will refer to this construction as the v + 2 u or the doubling construction for quadruple systems.) It is important to note that there need be no relationship between (X, x)
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and ( Y ,y ) , that F and G can be any 1-factorizations, and that a can be any permutation. We will also denote this quadruple system by [ X U Y ] ( x ,y , F, G, a). The following two observations are necessary for what follows: First, if (S, T) is a quadruple system containing the subsystem ( P , Q,) and ( P , Q,) is any quadruple system, then ( S , ( T \ Q,) U Qz)is a quadruple system. As with block designs this is referred to as “unplugging” the subsystem (P, Q,) and “plugging in” (P, Q,). As a consequence, if ( P , , Q,), (P2, Q 2 ) , . . . , (PI,Q,) are subsystems of ( S , T ) with [Pin P i ( S 2 and ( P , , Qf), (P,, QT), . . . , (P,, Q:) are any quadruple systems then
is a quadruple system. Secondly, a partial 1-factorization of the set X is a set F* = {G,F:, . . . ,FC} where each FT is a collection of pairwise disjoint 2-subsets of X and such that FTnFT [email protected] partial 1-factorization F* of X is said to be embedded in the 1-factorization F of Y if and only if X E Y and each partial 1-factor of F* is contained in a 1-factor of F such that the 1-factors containing FT and FT are different whenever i # j . In [2] A. Cruse has shown that if F* is a partial 1-factorization of X where 1 x1 is even, then the partial 1-factorization F* of X can be embedded in a I-factorization F of Y for every Y such that IYI is even and 2 2 1x1.This is awfully important in what follows.
Lemma 4.1. Let ( X , A ) be any quadruple system and {a, b, c, d } any block in A. Then ( X , A \{{a, b, c, d}}) can be embedded in a quadruple system ( X * , A*) such that {a, b, c, d * } A~* with d*E X*\X.
1x1= IYI. Let { a * . b*, c*. d*} be any block in B, and embed the partial 1-factorizations Proof. Let ( Y ,R ) be any quadruple system with X n Y = $3 and F* ={{[a.bl, Cc, d B , { [ a , c l , [b, 0 , {[a,d l , [b, CIH, and
G* ={{[a*,b*],[c*. d*l), {[a*,c*],[b*, d*B, {[a*,d*]. [b*, d*l)}
x1= (YI3 into I-factorizations F of X and G of Y respectively. (We can assume 1 8.) If a is any pairing of the 1-factors of F and G such that the 1-factors {[a,b]. [c, dl) and {[a*,b*], [c*, d * n ; {[a. c ] , [b, d n and containing {[a*,c*],[b*,d * n ; and {[a,d ] ,[b, cl) and {[a*,d*],[b*, c*]} are paired, then the quadruple system [X U Y ] ( A ,B, F, G, a) contains a subsystem (P, Q) where P = {a. b, c, d, a * . b*. c * , d*} and Q and A having exactly the block { a , b, c, d } in common. If we unplug ( P , 0)and replace it with the quadruple system obtained from ( P , 0)by interchanging d and d* in the blocks of 0 then the only block in A that is affected is {a, b, c, d } and it is replaced with {a, b, c, d*} where, of course, d * $ X. Theorem 4.2 (B. Ganter [63. A partial quadruple system can always be embedded in a quadruple system.
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Proof. Let (P, Q ) be a partial quadruple system. The proof is by induction on JP), the order of the quadruple system. If )PI = 0 there is nothing to prove. So, let IPJ3 1 and let d be any element in P. Let Q ( d ) be the set of all blocks in Q containing d and let {al, b,, c,, d } , {a2, b,, c2, d } , . . . ,{ak,bk, c k , d } be a distinct listing of the blocks in Q ( d ) . By the induction hypothesis (P\{d}, Q\Q(d)) can be embedded in a quadruple system (S, T). Repeated use of Lemma 4.1 beginning with (S, T) gives a quadruple system ( X ,A ) containing (P\{d}, Q\Q(d)) and such A d,# P and that if {a,, b,, c,, d } Q ~( d ) there is a block s, ={a,, b,, c,, ~ , } E with d, # d, whenever i # j . It is important to note that Is, f l s, I s 1. Let (Y, B) be any quadruple system such that X f l Y = 9 and 1 x1= IYI. We can assume IYI is large enough so that B contains k blocks,:s s z , . . . ,s: which pairwise intersect in the element d (since d $ P\{d} we can assume d E Y). Let S, = {F,,,F,,, F3,} be any 1-factorization of s, and ST ={GI,, G2,, G3,} any 1-factorization of s:. Since Is, f l s , l s 1 and Is: f l s T l = 1, S = S, U S 2 U . . . U Sk and S* = ST US:U. . . US: are partial 1-factorizations of X and Y. Embed (X, A ) and (Y, B) in quadruple systems ( X * , A*) and ( Y * ,B*) where X* n Y*= @ and \X*l= lY*I = 2 1 x1(by the doubling construction) and embed S and S* into 1-factorizations F and G on X * and Y*. Let a be any pairing of the 1-factors of F and G such that the 1-factor containing the edges in F,, is paired with the 1-factor containing the edges in GI, (i = 1 , 2 , 3 ; = 1 , 2 , . . . , k ) . Then the quadruple system [X* U Y*] x (A*, B*,F, G, a) contains the 8-element subsystems (s, U sT, q , ) , (s2U sf, q2),. . . , (sk U s:, q k ) such that I(s, U sT) f l (s, U s:)1<2. Hence if we unplug each (s, U sT, 4 , ) and replace it with the quadruple system obtained by interchanging d, and d in the blocks of q, we obtain a quadruple system (p, containing the blocks in Q ( d ) . Since Q \ Q ( d ) s A * \ { s , , s2,. . . ,s k } and the only blocks of A* that are changed in the transformation from [X*U Y*](A*,B*,F, G 0 ) to (p,Q) are the blocks s,, s,, . . . , sk it follows that the partial quadruple system (P, 0 )is embedded in (p,6 )completing the proof.
a)
5. Small embeddings for partial triple systems With the fact in hand that any partial block design or quadruple system can always be embedded, a very natural question to ask is: just how small can the containing block design or quadruple system be? All of the embeddings given in Sections 2, 3, and 4 are quite large since among other reasons, all of the proofs involve induction. In this section we give a survey of small embeddings for partial triple systems. To date, as far as the author can tell, there are no small embeddings for partial S(2, k, u ) block designs with k 2 4 or for partial quadruple systems. It is an understatement to say that much remains to be done in this area. In 1975 the author [17] proved that any partial triple system of order u can always be embedded in a triple system of order 6 u + 3. Subsequently, this result was improved to 6u + 1 (unpublished). Quite recently I.D. Andersen, A.J.W.
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Hilton, and E. Mendelsohn [15], have shown that a partial triple system of order n can always be embedded in a triple system of order u for every u 2 4 n + 1 and (of course) u = l or 3(mod6). Unfortunately, the proof of this last result is extremely technical and requires something like 25 pages (typewritten). For this reason we content ourselves here with the u + 6 u + 3 and u -+ 6 u + 1 embeddings. These embeddings are based on a very well-known construction due to R.C. Bose [ I ] and a slight modification of a somewhat less well-known construction due to Th. Skolem [31].
Bose’s constr~ctioo.Let (Q,.) be an idempotent commutative quasigroup of order 2u + 1; i.e., a quasigroup satisfying the identities x2 = x and xy = yx. We can assume Q = { 1 , 2 , 3 , . . . , 2 u + 1). Now set S = Q x { 1,2,3} and define a collection of triples t of S as follows: ( l j {(x, l ) , (x, 2), (x, 3 ) ) t~for every x E 0; and (2) if x f y, the three triples {(x, l), (y, 11, (xoy, 2)), {(x, 2), (y, 21, ( X O Y , 311, and {(x, 3 ) , (Y. 31, (XOY, 1))E t. It is routine to see that ( S , t) is a triple system.
Skdem’s coustructioo. In what follows we will call a quasigroup (Q, 0 ) halfidempotent and say that it satisfies the identity i x 2 = x if and only if I Q J = 2 u , Q = { l , 2 , 3 , . . . ,2u}, and
It is a trivial exercise to see that a half-indempotent commutative quasigroup of order u exists if and only if u is even. So let (0,0 ) be a half-idempotent commutative quasigroup of order 2u and set S = (Q x { 1,2,3}) U{m}, where m is a symbol which does not belong t o Q x { 1,2,3}. Define a collection of triples t of S as follows: (1) {(x, l), (x,2) ( x , 3 ) } ~ tfor every X E Qand X S U ; (2) for each x > u, the three triples {a. (x, l ) , (x - u, 2)}, {a, (x, 2), (x - u, 3)}, and {a,(x, 3 ) , (x - u, 1 ) ) t;~ and (3) if x f Y, the three triples {(x, I), (Y, I), b o y , 211, {(x, 21, (y, 2), ( X O Y 311, , and {(x, 3)}, (y, 3), (xoy, 1 ) ) 1.~ As with the Bose Construction, it is straightforward to see that ( S , 1 ) is a triple system.
Remarks. The original Skolem Construction uses an abelian group instead of a half-idempotent commutative quasigroup. The substitution of a half-idempotent commutative quasigroup not only simplifies the Skolem Construction (in the author’s opinion) but is absolutely necessary for the u + 6 u + 1 embedding to follow. It is also worth noting that the combination of the Bose and Skolem Constructions give an incredibly simple (and easily understandable) proof of the existence of a Steiner triple system of every order u = 1 or 3 (mod 6).
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We will need the following two results in order to undertake the 6v + 3 and + 1 embedding. The fkst is due to Allen Cruse and is stated without proof. It is easier to state and prove these results in terms of latin squares than in terms of quasigroups. 6v
Theorem 5.1. (A. Cruse [2]).A partial n x n idempotent commutative latin square can be embedded in a t x t idempotent commutative latin square for every odd t 2 2n + 1 and a t X t commutative latin square for every t 3 2n. Lemma 5.2. A partial idempotent commutative latin square of order m can be
embedded in a half-idempotent commutative latin square of order 2n for every n 5 m.
Proof. Let P be a partial m X m idempotent commutative latin square and define a partial n x n latin square P as below.
P=
Now embed 5.1.
p is based on 1 , 2 , . . . ,n and the only cells outside of P which are occupied are on the main diagonal and they are occupied by m + 1, m + 2 , . . . , n in that order.
in a 2n X 2n commutative latin square M by Cruse's Theorem
Since the order of M is even ( = 2n) each symbol occurs on the main diagonal of M an even number of times. Since each of 1,2, . . . , n already occurs on the main diagonal of M in P,,each must also occur on the main diagonal of M outside of P. A suitable permutation of the rows and columns n + 1, n +2, . . . ,2n transforms M into a half-idempotent commutative latin square containing P.
Theorem 5.3. [17]. A partial triple system of order v can always be embedded in a triple system of order 6k + 3 for every k 3 v.
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Proof. Let ( P , p) be a partial triple system of order v and define a partial binary operation on P as follows: (1) x o x = x for all X E P , and (2) if x # y, x 0 y is defined and x y = z if and only if {x, y, z} is a triple of p. It is routine to see that is well defined and that (P,") is a partial idempotent commutative quasigroup; i.e., xox = x for all x E P, and whenever x o y is defined then so is y o x , and furthermore xoy = yox. By Cruse's Theorem 5.1 we can embed ( P , in an idempotent commutative quasigroup of order 2k + 1 for every k B u. So let (Q, 0) be an idempotent commutative quasigroup of order 2k + 1 containing the partial quasigroup ( P ,0 ) . Set S = Q x {1,2,3} and define a collection of triples t on S by modifying the Bose Construction as follows: (1) {(x, l), (x, 2), (x, 3 ) ) t~for every x E Q; (2) for each triple {x, y, z } E ~ ,the three triples {(x, l), (y, l), (z, l)}, {(x, 2), (y, 2). (z,2)1, and {(x, 3, (Y,3), (2, 3 ) ) t~; (3) for every triple {x, y, Z ) E p , the six triples listed below belong to t : 0
0
0
0 )
{(x. I). (y. 2), ( 2 . 3 ) )
{(x, 2 ) , (Y, 3). (2, 1))
{cx, 1). (Y, 3, (z,2)}
{(x. 3), (Y. 11, (272))
{(x, 2). (Y. 11, (2.3))
{(x, 3), (Y, 2),
(2, 1));
and
(4) if x # y and x and y d o not belong t o a triple of p the three triples {(x, I), ( y . l), ( x O y , 2)). {(x, 2), (Y. 2). ( x ~ Y3)). , and {(x, 3), (Y, 3), boy, 1 ) ) t.~ Part (2) of this construction guarantees that 3 disjoint copies of ( P , p) are embedded in (S, t ) and so it only remains to show that ( S , t ) is a triple system. To see that (S, t ) is indeed a triple system it suffices to show that every pair of distinct elements of S belong t o at least one triple of r and that I t \ G ( S ((IS]- 1)/6= $(6k + 3)(6k + 2 ) . It is trivial to see that every pair of distinct elements of S belong to at least one triple of t. On the other hand, by direct count, there are
2k-t 1 triples of type (I), 3 Ip1 triples of type (2). 6 ( p ( triples of type ( 3 ) , and 3[(2k + 1)(2k)/2- 3 lpl] triples of type (4). Since the sum of these numbers is i(6k + 3)(6k + 2 ) it follows that (S. t ) is a triple system.
Tbeorem 5.4. A partial triple system of order v can always be embedded in a triple system of order 6k + 1 for every k 3 u. Proof. Let (P, p) be a partial triple system and let (P, be the partial idempotent commutative quasigroup associated with (P, p). Since k 3 u we can embed ( P , in a half-idempotent commutative quasigroup (Q, 0) of order 2k by Lemma 5.2. Now set S = (0x { 1,2,3))U{m} and modify the Skolem Construction to obtain 0)
0)
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the following collection of triples: (1) {(x, l), (x, 2), (x, 3 ) ) t~for every x E Q and x < k ; (2) for each x > k , the three triples {CQ, (x, l), ( x - k , 2)), {a,(x, 2), (x- k, 3)}, and {m, (x, 3), (x - k, 1 ) )t ~ ; (3) for every triple { x , y, Z } E p, the three triples {(x, l), (y, l), (T,l)}, {(x, 2), (Y, 21, (z, 2)}, and {(x, 31, (Y, 3), (Z93)IE t ; (4) for every triple {x, y, Z } E p, the six triples listed below belong to t :
{(x, I), (Y, 21, (z,3)1
{(x, 3, (Y, 31, (z, 1))
{(x, I), (Y, 3), (z, 2))
{(x, 3), (Y, I), ( z , 2 ) }
{(x, 21, (Y, I), (z, 3))
{(x, 3), (Y, 2), (z, 1)); and
( 5 ) If x # y and x and y do not belong to a triple of p, the three triples {(x, l), {(x, 21, (Y, 2), b o y , 3))- and {(x, 31, (Y, 31, (XOY, 1 ) ) E t. (Y, 11, (XOY, Part (3) guarantees that, as was the case in the u + 6 u + 3 embedding, 3 disjoint copies of (P, p) are embedded in (S, t ) . It is trivial to see that every pair of elements of S belong to at least one triple of t and so to complete the proof we must show that ltISi(6k + 1)(6k). By direct count there are:
k triples of type (l), 3k triples of type ( 2 ) , 3 IpI triples of type (3), 6 ) p l triples of type (4), and 3[(2k)(2k- 1)/2-3 triples of type ( 5 ) .
[PI]
A simple calculation shows that the sum of these numbers is i(6k + 1)(6k) which completes the proof. Combining Theorems 5.3 and 5.4 gives the following theorem.
Theorem 5.5. A partial triple system of order n can always be embedded in a triple system of order u for all u Z 6n + 1 and u 3 1 or 3 (mod 6). We close this section by stating the Theorem of Anderson, Hilton, and E. Mendelsohn which is the best result to date with respect to the size of the containing triple system. As was previously mentioned, the length (and complexity) of the proof makes it prohibitive to reproduce here.
Theorem 5.6. (I.D. Andersen, A.J.W. Hilton, and E. Mendelsohn [153. A partial tiple system of order n can always be embedded in a tiple system of order u for every u a 4 n + 1 and (of course) u = 1 or 3 (mod 6). 6. Doyen and Wilson’s theorem It is an easy exercise to see that if ( S , t ) is a triple system and (P, p) is a subsystem of (S, t ) , then IS1 2 2 /PI+ 1. A quite natural question to ask then is: For
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a given triple system (P,p), for which u 2 ]PI + 1 ( u = 1 o r 3 (mod 6) of course) does there exist a triple system (S, t) containing ( P , p) as a subsystem. In 1973 J. Doyen and R.M.Wilson [4] showed that any triple system of order u can always be embedded in a triple system of order u for every u 2u + 1. We use the Bose and Skolem constructions to give a somewhat different proof than the original proof given by Doyen and Wilson. As was the case with Lemmas 5.1 and 5.2, it is easier to state and prove the following lemma in terms of latin squares than in terms of quasigroups. In what follows we will say that the 2n x 2n half-idempotent commutative latin square N based on 1,2, . . . , 2 n contains the 2m x 2m half-idempotent commutative latin subsquare M if and only if M is based on 1 , 2 , . . . , rn; n + 1, n + 2 , . . . , n + m and is situated in N as indicated in the accompanying picture.
M occupies the shaded cells and is based on 1 . 2 , . . . , r n ; n + l , n + 2 ,..., n + m .
Lemma 6.1. For every n 2m + 1 such that n and m haue the same parity there exists a 2n x 2n half-idempotent cornmutative latin square containing a 2m x 2m
half-idempotent commutative latin subsquare.
Proof. Let A and B be the latin squares defined below.
n xn commutative latin square based on (1.2 , . . . , m } U { n + m + l , . . . , 2 n } where P is an m x rn latin subsquare based o n 1 , 2 . . . . , rn.
A=
n x n latinsquare b a s e d o n { t n + l , r n + 2 , . . . . n}U ( n 1,. , , , n m } where Q is an m x rn latin subsquare based on n + 1.. . . , n + m and the main
+
,
. *
+
diagonal outside of Q looks like m + 1 , rn + 2 , . . . n in that order. (Such a square is guaranteed by a result due to A.J.W. Hilton [14J).
.
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Now define M to be the 2n X 2n latin square given below.
M=
1
"'I&! '"
'
1
A
2n x 2n commutative latin square (BT is the transpose of B ) containing a 2m x 2m latin subsquare T (the shaded cells).
Now, if we unplug T and replace it with a 2m X 2m half-idempotent commutative latin square and, for each i = 1 , 2 , . . . , n - myinterchange m + i and x in the cells {(m+i, m + i ) , ( m + i , n + m + i ) , ( n + m + i , m + i ) , ( n + m + i , n + m + i ) } the resulting square M is a half-idempotent commutative latin square and, of course, contains a 2m x 2m half-idempotent commutative latin subsquare. Before beginning the proof of Doyen and Wilson's Theorem we remark that since triple systems have the replacement property, it is only necessary to construct for every u 2 2 u + 1 a triple system of order u containing a subsystem of order u.
Lemma 6.2. Any triple system of order u = 3 (mod 6) can always be embedded in a triple system of order u for every u 3 2u + 1. Proof. The proof splits into two parts. (1) u = 1 (mod 6). Write u = 6k 1. Since u 2 2 u + 1 it follows that 2k 22(fu). Further, since u SE 3 (mod 6), $u is an odd positive integer. Hence by Lemma 5.2 there exists a half-idempotent commutative quasigroup (0,0 ) of order 2k containing an idempotent commutative subquasigroup (P,") of order $. If we take Q ={l,2 , . . . , 2 k } and P ={l,2 , . . . ,u/3} then the Skolem Construction gives a triple system of order u containing a subsystem of order 3611) = u. (2) u=3(mod6). Write u = 6 k + 3 . Since u 2 2 u + l , 2 k + 1 > 2 6 u ) + l . By Cruse's Theorem 5.1 an idempotent commutative quasigroup of order n can always be embedded in an idempotent commutative quasigroup of order m for every odd m 2 2 n + 1. So let (Q, 0 ) be an idempotent commutative subquasigroup of order 2k + 1 containing an idempotent commutative subquasigroup of order fu. The Bose Construction then gives a triple system of order u containing a subsystem of order 3(fu) = u. Combining cases (1) and (2) completes the proof.
+
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We now concern ourselves with the problem of embedding triple systems of order L’ = 1 (mod 6).
Lemma 6.3. Any triple system of order u = 1 (mod 6) can always be embedded in a triple system of order u for every u 2 4 1 1+ 3. Proof. Since u = 1 (mod 6), 2u + 1 = 3 (mod 6). Hence by Lemma 6.2 there is a triple system of order u containing a subsystem of order 2 u + 1 for every u 3 2(2u + 1) + 1 = 4u + 3. Taking the subsystem of order 2u + 1 to have a subsystem of order u completes the proof. Now, if u = 1 (mod 6), 4 u + 3 = 1 (mod 6). Hence our proof will be complete if we can construct a triple system of order u containing a subsystem of order u whenever 2u + 1 < u S 4 u - 1. We achieve this in four steps. Lemmas 6.4, 6.5, and 6.6 are all due to Doyen and Wilson [4].
Lemma 6.4. [4]. Any triple system of order u = 1 (mod 6) can always be embedded in a triple system of order u for every 3u zs u s 4u - 1 . Proof. In [4] Doyen and Wilson construct for every u and u satisfying the above conditions a PBD (pairwise balanced design) ( P , B ) such that lPl= :(u - 1) and every block has size 3 0 or 1 (mod 3) and at least one block has size $(u - 1). Now modify the u + 2u + 1 construction for triple systems as follows: Let = be an element not in { 2 , 3 }x P, S = {m} U ( ( 2 . 3 )x P ) , and define a collection of triples f of s as follows: ( 1 ) The IPJtriples {=. (2,x), (3, X)}Et for every X E P ;and ( 2 ) For every block { x , . x 2 . . . . . x , , } e R define any triple system on {m}U ( { 2 , 3 }x {x,, x2. . . . . x,,}) with t h e proviso that the triples {t.,( 2 , x i ) , (3. x i ) } ; i = 1. 2. . . . . n ; belong to the collection, and place these triples in B. Part (2) of the above construction can in fact be carried out as a consequence of t h e fact that every block in B has size n G O or 1 (mod 3) and so 2n + 1= 1 or 3 (mod 6) which is the order of a triple system. Since ( P , B) contains a block of size i(c - 1). (S. t ) has a subsystem of order u and of course IS1 = u.
Lemma 6.5. [4]. A n y triple system of order u = 1 (mod 6) can always be embedded in u triple system of order u = 3 (mod 6) for every 2u + 1 u 3v. Proof. In 1251 R. Peltesohn has shown the existence of a cyclic triple system of every order except u =9. So. let (S, 1 ) be a cyclic triple system of order u = 1 (mod 6). If OL is any cyclic permutation o n S, then (a)has b ( u - 1) orbits of length u. Define r* t o be the union of any i ( u - 2 u - 1) of these orbits. Let x be any point of S and let P be the subset of all points of S (including x ) which belong to a triple of f* containing x. Since u = 3 (mod 6) and u = 1 (mod 6), (PI = u - 2 u is
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the order of a triple system. Now, let (P, p ) be any triple system of order u -2u, Q = P U (S x { 1,2}), and define a collection q of triples of Q as follows: (1) Every triple in p belongs to q ; (2) thetnple{a,(xa', 1 ) , ( a a i , 2 ) ) ~ q f o r e v e r y a ~ P a n d i = 0 ,,..., 1 , 2 u-1; (3) for every {a, b, C } E t* the two triples { ( a , l), (b, l), (c, 1)) and {(a,2), (b,2), (c, 2)) belong to q ; and (4) for every {a, b, C } E r \ t * the following four triples belong to q: { ( a , l), (b, l), (c, I)), { ( a , 11, (b, 3 , (c, a),{(a,a , (by I), (c,2)}, and { ( a , (b, 3 , (c, 1)). It is a routine matter to see that (Q, q ) is a triple system of order IQ(= 2u + ( u - 2u) = u and of course (Q, q) contains the subsystem (S, t ) of order u. Our proof will now be complete if we can show that a triple system of order u = 1 (mod 6) can always be embedded in a triple system of order u = 1 (mod 6) for every 2 u + l = z u < 3 u . We handle this in two parts: the first where u - u = 6 (mod 12) (Lemma 6.6) and the second where u - u = O (mod 12) (Lemma 6.7).
Lemma 6.6 [4]. Any triple system of order u = 1 (mod 6) can always be embedded in a triple system of order u =1 (mod 6), where 2 u + l S u < 3 u and u - u = 6 (mod 12). Proof. Let (S, t ) be any triple system of order u where S = {1,2,. . . ,u}. In [4] Doyen and Wilson construct a PBD (P, B ) of order u - u with the property that the blocks of B can be partitioned into subsets B*, B,,B,,. . . , B, where B* consists of blocks of size 3 and each Bi is a parallel class of blocks of size 2. Now set Q = S U P (we can assume SnP=p)) and for each i = 1 , 2 , ..., u set By= {{x, y , i} I {x, y } B~i } . If we now take q = B* U B; U * U BS U t it is immediate that (0, q) is a triple system of order u containing ( S , t ) as a subsystem.
-
Lemma 6.7. A n y triple system of order u = 1 (mod 6) can always be embedded in a triple system of order u = 1 (mod 6), where 2u + 1 u <3u and u - u = 0 (mod 12). Proof. Write u = 6 k + 1 . Since u s 2 u + 1 , k s 2 [ i ( u - l ) ] + 1 . Since u - u = O (mod 12), k and i ( u - 1) have the same parity, and so by Lemma 6.1 there exists a half-idempotent commutative quasigroup (Q, of order 2k containing a halfidempotent commutative subquasigroup (P, of order $(u - 1). The Skolem Construction now gives a triple system of order u with (because of the halfidempotent commutative subquasigroup (P, a subsystem of order 3[;(u - 1)]+ 1=u. 0 )
0)
0))
Combining all of the above lemmas gives us the following theorem.
Theorem 6.8. (J. Doyen and R.M. Wilson [4]). A n y triple system of order u can always be embedded in a triple system of order u for every u > 2u + 1.
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Remarka. The introduction of a 2n x 2n half-idempotent commutative latin square containing a 2 m x 2m half-idempotent commutative latin subsquare is not necessary for the proof of the Doyen and Wilson Theorem. It is used only in the proof of Lemma 6.7 which can be achieved by other means. There are two reasons for the inclusion of Lemma 6.1 in these notes: The first is that it is a new technique (as far as the author can tell) for constructing triple systems containing specified subsystems. The second is that the construction for all m and n such that n 3 2m + 1 of a 2n x 2n half-idempotent commutative latin square containing a 2m X 2 m half-idempotent commutative latin subsquare is an open (and interesting) problem. The solution of this problem for m even and n odd allows the amalgamation of Lemma 6.6 and Lemma 6.7 above. Indeed, such a solution would allow a very slick proof that any triple system of order u = l (mod6) can always be embedded in a triple system of order u for every u 2 2 u + 1 and u = 1 (mod 6).
7. Intersectioo peseming embeddings In 1977 Jon Hall and J.T. Udding [8] proved the following theorem.
Theorem 7.1. (J. Hall and J.T. Udding [8D.Let ( P , p ) be a partial m’ple system of order u. Then if either u = 3 (mod 6) and u 2 6 u + 3 or u = 1 (mod 6) and u 12u + 7, there exists a pair of triple systems ( S , tl) and (S, t2) of order u such that ( P , p ) is embedded in each of ( S , t l ) and (S, t 2 ) and furthermore t , f l t, = p. We begin this section by strengthening Theorem 7.1 as follows.
Tbeorem 7.2. Let (P, p l ) and ( P , p z ) be partial triple systems of order u. Then for every u 3 6 u + 1 there exists a pair of triple systems ( S , tl) and ( S , f 2 ) of order u such that ( P , p l ) is embedded in (S, t l ) , ( P , p 2 ) is embedded in ( S , t 2 ) , and 1, n t 2 = PI
nP 2 .
Proof. Let ( P , p l ) and ( P , p 2 ) be any pair of partial triple systems of order u and let u s 6 u + 1 . The proof, quite naturally, splits into two parts. The second part is due to Hall and Udding. ( 1 ) u = 1 (mod 6). Embed (P, p l ) into a triple system ( S , t l ) of order u exactly cs in Theorem 5.4. NOW,embed ( P , p,) into a triple system (S, t,) of order u by modifying parts (2) and (5) of the construction in Theorem 5.4 as follows: (2’) for each x > k, the three triples {m, (x, l), ( x - k, 3)}, {m, (x, 2 ) , (x - k, l)}, and { x , ( x , 3), ( x - k, 2 ) ) t ~; and (5’) if x f y and x and y do not belong to a triple of p2, the three triples {(x, 1). ( x , 1). (XOY,3% {(x, 2), ( Y , 21, ( X O Y , 1)), and {(x,31, (Y. 31, b o y , W E t. Now let a be the permutation on Q defined by i a = k + i if l s i s k , and ia = i - k (mod k)+ 1 if i > k. Define a permutation p on S by: /3 is the identity
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Embedding theorems for Steiner systems
mapping on (Q ~ { lU{a} } ) and (x, l ) p = (xa, i) for all (x, i ) E Q x{2,3}. Since 0 is the identity mapping on (Qx{l})U{a~},(P, p2) is still embedded in (S, t z P ) and a straight forward check shows that t , n t2@ = p1n p2. (2) u = 3 (mod 6). Embed (P, pl) into a triple system (S, t,) of order u exactly as in Theorem 5.3. Now, embed (P, p2) into a triple system (S, t,) of order u by replacing part (4) of the construction in Theorem 5.3 with: (4’)if x # y and x and y do not belong to a triple of p,, the three triples {(x, l), t. (y, l),b o y , 311, {(x, 21, (Y, 21, b o y , I)}, and {(x, 31, (Y, 31, b o y , Let a be any permutation on Q which fixes no points of Q and such that Pa r l P = 9 (this is possible since IQl5 2 \PI + 1). Define a permutation p on S by: p is the identity mapping on Q X (1) and (x, i ) p = (xu, i ) for all (x, i) E Q X {2,3}. As with the first part, since p is the identity mapping on Q ~ { l }( ,P , p2) is embedded in (S, t2p) and it is routine to see that 4 n t2@ = p1 np,. Let ( P , pl), ( P , p2), . . . ,(P, pk) be any collection of partial triple systems. A collection of triple systems (S, tl), (S, t2), . . . ,(S, tk) such that (P, pi) is embedded in (S, ti) and such that 6 fl ti = pi f l pi for all i # j is called an intersection preserving set of triple systems for the partial triple systems (P, pl), (P, p2), . . . ,(P, pk) [23]. Theorem 7.2 says that any pair of partial triple systems of order u can always be embedded in an intersection preserving pair of triple systems of order 6u + 1. An obvious question to ask at this point is whether or not an arbitrary collection of partial triple systems can always be embedded in an intersection preserving set of triple systems? In 1975 the author and A. Rosa [23] showed that this is always possible. However, the construction given in [23] gives rather large containing triple systems. Subsequently a much smaller (but still far from best possible) embedding was obtained in [22]. We give here a slight variation of this latter embedding. The following definition may seem a bit “far out” at first, but as we shall see, it is exactly what is needed in what follows. A set of 2k n X n latin squares Q(i, j); i E {1,2}, j E {2,3, . . . , k}; is said to be a separating set of latin squares of order n and strength k provided that Q ( l , i) and O(1, j ) , i f j , agree in exactly the one cell in their upper left hand corners (cell (1, I)) and each Q(2, i) is disjoint from all of the other latin squares except possibly Q(1, i). (Two latin squares are disjoint provided they do not agree in any cell.) For example, the 4 latin squares given below are a separating set of order 5 and strength 2.
r
7 3 3 4 4 5 5 1
3 4 5 1 2
4 5 1 2 3
5 1 2 3 4
A 3 4 5 2 ) 1 3 4 5
4 2 5 3
2 5 1 4
5 1 3 2
Q(2, 1 )
3 4 2 1
4 5 1 3 2
5 1 3 2 4
1 3 2 4 s
3 2 4 5 1
Q(2.2)
2 4 5 1 3
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The following result (stated without proof) is the main tool used in showing that any collection of partial triple systems can always be embedded in an intersection preserving set of triple systems.
a separating set of latin squares of order n and strength k for every n 2 2 k + 1 such that n 2 or 4 (mod 6).
Lemma 7.3 [20]. For every positive integer k there exists
Theorem 7.4. Any collection of k partial tiple system of order m can always be embedded in an intersection preserving set of tiple systems of order M ( n - 1 ) + 1 for every M 2 4 m + 1 and M 1 or 3 (mod 6), and for every n = 3 (mod 6 ) and n>2k+1.
Proof. Let ( P . p l ) , (P, p2), . . . ,( P , p k ) be any collection of partial triple systems of order m. Use Theorem 5.6 to embed each ( P , p,) in a triple system (S, f , ) of order M. Unfortunately this process of embedding the partial triple systems (P, p l ) , ( P , p 2 ) , . . . ,( P , p k ) does not necessarily preserve intersection. That is, it can happen (and usually does) that p, n p, c r, fl tl where inclusion is proper. We rectify this situation by “straining out” the undesirable triples via the use of a separating set of latin squares. Let Q(i, j ) ; i E { 1 , 2 } , j E { 1 , 2 , . . . , k}; be a separating set of latin squares of order n - 1 and strength k where n = 3 (mod 6 ) and n > 2 k + 1 (with cell (1, 1) in each Q( 1, i ) occupied by 1) and denote by (Q,@(i,j)) the quasigroup corresponding to Q(i, j). Additionally set Q* = Q U{m} where is a symbol not in Q, and let (Q*, q l ) , (Q*, q2), . . . , (Q*, qk) be k mutually disjoint triple systems of order n. (In [3] J. Doyen proved that the number of mutually disjoint triple systems of order n = 6r + 3 is at least 41 - 1.) Let R = {m}U (Q x S) and form the singular direct products ( R , T , ) .( R , T2),. . . ,( R , Tk) as follows: (1) {m. (b. x ) , (c, X ) } E T, for every {m, b, c } E ~ and , XES, ( 2 ) {(a,x), (b, x ) , (c, X ) } E T, for every {a, 6, C}E qi, and where m${a, b, c } and x E S. and ( 3 ) {(a,XI,(b, y), (a@,b,Z))E T,, where a, b e Q, {x, y, Z)E 4, x < y ( 2 , and @ , = @ ( l , i ) if { x . y , z } ~ p ,and @ , = @ ( 2 , i ) i f { x , y , ~ ) ~ f , \ p , . It is a routine matter to see that each (R, T,) is a triple system of order M(n-1)+ 1 . We now show that the triple systems (R, T,), (R, T2),. . . ,(R, Tk) are an intersection preserving set of triple systems for the partial triple systems (P, P I ) , (P, P A . . . (P, Pk). To begin with, suppose {a,b, c} E pi and let x < y < z be the order of a, b, c. Then by definition ((1, x), ( 1 , y ) . ( 1 ‘8,1 , Z)}E T,, where 1 1 = 1@(1. i)l = 1 (since { x ,y. ZIE p , ) . Hence ((1,x ) , ( 1 , Y), (1, z)}={(l, a ) , (1, b), (1, C ) } E T and so if we identify each element x in P with (1, x) in R , then {a,b, C}E p, if and only if ((1, a ) , (1, b ) , ( 1 , C)}E T,. It follows that (P, p,) is embedded in (R, T,) and, of course. p1n p, G T, fl T,. We now show that, in fact, p, f l p, = TIn T,. Now. since (Q*,4,) and (Q*. q ) are disjoint, (R, T,) and (R, T,) cannot have (x:
7
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any triple of type (1) or (2) in common. Hence the only possible triples that ( R , T i )and (R, q)can have in common other than the triples in pi npi are of the form {(a,x ) , (b, y ) , (c, z)}, where {x, y , Z } E (6 r l 4)\(pi npj).We can assume {x, y , z } $ pi and that x < y < z. Let a, b E Q and consider the triples {(a,x), (b, y ) , ( a @ b, z)} in Ti and {(a,x), (b, y), b, 2)) in q.Since {x, y , z}$ pi, we have by definition that a b = a @(2,j ) b, so that regardless of whether a @i b = a @( 1, i)b or a@ib = a@(2, i)b we still have a B i b# a@jb. It follows that Ti and have exactly the triples ( ( 1 ,x ) , (1, y ) ( 1 , z ) } in common, where (x, y , Z}E pi np,. In 1977 in a series of two papers [26,27] W.B. Poucher obtained an analogue of Theorem 7.4 for block designs in general. We give a brief sketch of Poucher’s work here and refer the reader to [26,27] for the lurid details. We will not concern ourselves with size in what follows since everything is extremely large (just how large is not certain since the existence of the embeddings is based on induction). As was the case with triple systems a collection of block designs (X, xl), (X, x,), . . . ,(X, q) is an intersection preserving set of block designs for the partial block designs (P, p l ) , (P,p2), . . . ,(P, p,) provided (P, p i ) is embedded in (X, x i ) and 4 n3 = pi npi. An n 2 x k orthogonal array is a pair (A,R) where A ={l, 2,. . . , n } and R is a collection of k-tuples of elements of A (called rows) such that if i < j E {1,2,. . . , k } and x and y are any two elements of A (not necessarily distinct) there is exactly one row in R whose ith coordinate is x and whose jth coordinate is y. A set of 2 r n Z x k orthogonal arrays (A,R(i, j)); i E { l , 2 } , j E { 1 , 2 , . . . , t}; is called an OA(t, k, n ) separating set of orthogonal arrays provided that (A,R(1, i ) ) and (A,R(1, j)) have exactly the one row (1, 1, 1, . . . , l)(all 1’s) in common and each ( A ,R ( 2 , i ) ) has no row in common with any of the other orthogonal arrays except possibly (A, R(1, i)). By a E(r, k, v ) embedding package is meant a collection of t S(2, k, u ) pairwise disjoint block designs and an OA(t,k, u ) separating set of orthogonal arrays. The following result due to W.B. Poucher is the main tool used in what follows.
aj
(aai
Lemma 7.5. (W.B. Poucher [26,27J. For every pair of positive integers t S 2 and k 3 3, there exists an infinite set of positive integers V such that v E V if and only if a E(t, k, u ) embedding package exists. The proof of the following theorem is now quite easy and is similar to proof of Theorem 7.4.
Theorem 7.6. (W.B. Poucher [26,27]. Any collection of partial block designs can always be embedded in an intersection preserving set of block designs.
Proof. Let (P, p l ) , (P, p2), . . . ,(P, p,) be a collection of partial block designs with block size k. Use Ganter’s Theorem 3.4 to embed each (P, p i ) in a block design
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( X , . x , ) . By taking direct products and suitable labelings we can assume that X I = X, = . . . = XI = X . As with the case for triple systems there is n o guarantee that the block designs (X, x l ) , (X, x2), . . . , (X, x,) are an intersection preserving set of block designs for the partial block designs (P, p,), (P, p,), . . . , (P, pt). So, let E(r, k, u ) be an embedding package where (Q,q , ) , (Q,q2).. . . , (Q,q l ) are pairwise disjoint S(2, k, u ) block designs and (0, R ( i , j)); i E {1,2}, j E {1,2, . . . , ? } ; are an OA(t,k, u ) separating set of orthogonal arrays. Set S = Q x X and for each i = 1.2,. . . , t define a collection T, of blocks of S as follows: (1) { ( a l , c ) , ( a 2 , c ) , . . , ( a , . c ) } ~ T ,for every { a , , a 2 , . . , ak}Eq, and C E X ; and ( 2 ) { ( a l - c , ) ,(a*,C,),...,(ak,Ck)}ET,; where{c,,c,,...,Ck)EX;, C , < c 2 < . ’ ’ < ck, and ( a , ,a 2 , . . . , U k ) E R(1, i ) if {Cl, C2,. . . ,ck}€ p, and (al, a,, . . . , a,)€ R(2, i) if { c , .c,, . . . , c k } cx, \p,. ’The proof that (S, TI),(S, TI), . . . , (S, TI)is an intersection preserving set of block designs for the partial block designs (P, p,), (P, p2), . . . ,(P, pk) is similar to the proof of Theorem 7.4 and so is omitted here. The reader is referred to [26,27] for details. We finish up this section by showing that quadruple systems have the intersection preserving embeddability property. The technique of embedding is right in line with the techniques used for block designs and so we give a very brief outline here and refer the reader to [24] for details. A separating set of 3-quasigroups of order u and strength t is simply a collection of 2t 3-quasigroups (0, ( , , ),,); i E {1,2}, j E { 1 , 2 , . . . , t ) ; such that (1, 1, 1)1,= ( l ,1, = 1 for all i, j E { l , 2 , . . . , t } ; (a, b , ~ ) , ~ f b( ,a ~, ) i~# j,, , for all a, b, c E except a = b = c = 1; and (a, b, C)Zk # (a, b, c),, for all i , j except possibly i = 1 and j = k. By an EQ(t, u ) embedding package for quadruple systems is meant a collection of t pairwise disjoint quadruple systems of order u and a separating set of 3-quasigroups of order u and strength t. In [24] it is shown for a fixed t there are an infinite number of u for which an EQ(r, u ) embedding package exists.
Theorem 7.7. (C.C. Lindner and A. Rosa [24]). Any collection of partial quadruple systems can always be embedded in an intersection preserving set of quadruple systems. Proof. Let (P, p,), (P, p 2 ) , . . ,( P , p , ) be any collection of partial quadruple systems. By Ganter’s Theorem 4.2, and taking suitable direct products, these partial quadruple systems can be embedded in quadruple systems (V, u , ) , (V, u2),. . . , (V.u,). Let EQ(t,q) be an embedding package where: q > t ; ( Q , q l ) , (Q, q2), . . . , (Q,q l ) are pairwise disjoint quadruple systems of order q ; and (Q.( . , ) , I ) , i ~ { 1 , 2 } j, E { l , 2 , . . . , t } , is a separating set of 3-quasigroups of order q and strength t. Finally, let F={Fl,F,, . . . F, .,} be any 1-factorization of Q and a , , a2,. . . ,a, any r permutations on {1,2, . . . , q - 1) which d o not agree anywhere. (This is
.
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possible since 4 - 15 t.) Now form the generalized direct products (Q x V, T1), (Q x V, T2),. . . , (Q x V, TI)as follows: (1) { ( a ,w ) , (b, w ) , (c, w ) , (4 W ) } E Ti for every {a, b, C, d } qi~ and W E V; (2) for every pair of distinct elements u, w E V place {(x, u ) , (y, u ) , ( z , w ) , ( t , w ) } in Ti where [x, y] E F,, [z, t ] Fd, ~ and car, = d ; and (3) {(a,XI, (b, Y), (c, 21, ((a,b, c)i, W ) } E Ti where {x, Y, 2, W } E ui, x < Y < z < W , if {x, y, z, w } € p i and (a, b, cIi = ( a ,b, cIzi if {x, y, z, W}E and (a, b, c ) ~= ( a , b, ui \pi* The proof that (Q x V, Tl), (Q x V, Tz),. . . ,(Q x V, T,) is an intersection preserving set of quadruple systems for the partial quadruple systems ( P , p l ) , (P,p 2 ) , . . . , (P, p , ) (while tedious) is straightforward and can be found in [24].
8. Resolvable embeddings If (S, t ) is a S(2, k, u ) block design and T is a collection of blocks which partition S, then T is called a parallel class of blocks of (S, t ) . A block design whose blocks can be partitioned into parallel classes is said to be resolvable. An obvious necessary condition for the existence of a resolvable S(2, k, u ) block design, in addition to u-l=O(mod k(k-1)) and u(u-l)=O(rnod k(k-1)) (see Section 3), is u = O (mod k). These necessary conditions for the existence of a resolvable S(2, k, u ) block design are also sufficient for k = 3 (Kirkman triple systems) (D.K. Ray-Chaudhuri and R.M. Wilson [293, k = 4 (D.K. RayChaudhuri, R.M. Wilson, and H. Hanani [13J), and for sufficiently large u (D.K. Ray-Chaudhuri and R.M. Wilson [303. The obvious question to ask here is whether or not a partial block design can always be embedded in a resolvable block design. If we do not mind the containing block design being rather large, the answer is yes. In fact, we can impose some interesting additional requirements on the embedding and still embed any partial block design in a resolvable block design. In what follows we will not be concerned with size since all of the embeddings are based on Ganter’s Theorem 3.4 and 4.2 (and so are awfully large). By a partial parallel class of blocks of the (partial) S ( 2 , k, u ) block design (P, B) is meant a collection of pairwise disjoint blocks of B. Now let (P, B) be a (partial) block design TB ={rl, rz,. . . ,rt}a resolution of the blocks of B into partial parallel classes, and (S, T) a resolvable block design containing (P, B), i.e., (P, B) is embedded in (S, T). Furthermore, suppose there is a resolution ?rT of the blocks of T into parallel classes so that the parallel classes containing mi and ri are different whenever i# j . In this case, the resolution TT of T is said to preserve the partial parallel classes TB of B. In 1976 the author [18] proved that any partial block design (P, B) along with any partition of B into partial parallel classes can always be embedded in a resolvable block design which preserves these partial parallel classes. The proof is
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not difficult and relies on the following use of the u 4 (k - l ) u + 1 construction to construct resolvable block designs. Let k and t be positive integers such that there is a resolvable S ( 2 , k, ( k - 1 ) t + 1) block design (R, r ) and let (S, T) be any block design with block size 1. Now plug ( R ,r ) into the u + ( k - 1)u + 1 construction to construct from ( S , T ) a S ( 2 , k , ( k - 1)v + 1) block design ( K , B). For each x E S let b x , ,b x 2 , . . , b,, be the blocks in T containing x and denote by B,, the set of all blocks in the parallel class of {=, (2, x). (3, x), . . . ,( k , x)} in the parallel class of (R, r) generated by bxi (i.e., defined on {m}U ({2,3,. . . , k } x b x i )other than {m, (2, x). (3, x), . . . , ( k , x)}. Set 7 ~ ( ~ ) = { { 2 , ( 2( , 3~ ,) ~ , ). .~.(~,x)}}UB,IUB,ZU...UB,. .
Clearly T ( X ) is a parallel class of ( K , B) and { ~ ( x 1)all x E S} is a resolution of the blocks of B into parallel classes. The proof of the following theorem now goes quite quickly.
Theorem 8.1 [18]. Any partial block design ( P , B ) along with any partition of B into parallel classes can always be embedded in a resolvable block design which preserved these partial parallel classes.
Proof. Let ( P , B) be a partial block design with block size k and let T B = any partition of B into partial parallel classes. Additionally, let ( R , r) be any resolvable S(2, k, ( k - 1 ) t + 1 ) block design (see remarks at the beginning of this section) and set s = t - k - 1. For each block b E B let X ( b ) be a set of s symbols none of which belongs to P and such that if b, # b , B, ~ X ( b , )fl X ( b J = $4. We can further assume that none of the numbers 1 , 2 , 3 , . . . , n belong to P or to any X ( b ) . For each b E B set 6 = { i } U b U X ( b ) if and only if b E r , . Now take S to be the union of P and all of the & ( b E B ) and T = {6 1 b E B}.Of course (S, T ) is a partial block design with block size t and so we can use Ganter's Theorem 3.4 to embed (S, T) in a S ( 2 , t, u ) block design ( S * , T*). Now plug the S(2, k, ( k - l ) t + 1 ) resolvable block design (R, r ) into the u -+( k - 1 ) u + 1 construction to construct from ( S * , T*) a resolvable S(2, k. ( k I)r. + 1) block design (P*, B*) as in the remarks immediately preceding the statement of the theorem. Trivially if b = { y , , y , . . . . , y k } € r, we can require {(2. y,). (2, yz), . . . ,( 2 , y , ) } not only to be a block in the copy of (R, r ) defined on { x }U((2.3.. . . . k}x 6) but also to be in the same parallel class as {a. (2. i ) , (3, i ) , . . . , ( k . i ) } . If for each y E P we rename (2, y ) by y, then the original partial block design (P. €3) is embedded in (P*. B*) and, moreover, if b , and b Z E r,,then by construction both b , and b2 are in the parallel class r(i)of ( P * . B*) containing {s, (2, i), (3, i ) , . . . ,( k , i ) } . Hence the resolution TB* = { d x ) 1 all x E S*} is a resolution of ( P * , B*) which preserves the partial parallel classes T B of B, completing the proof. {T,, T?,. . . , T,) be
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One drawback of the embedding given by Theorem 8.1 is that it depends upon the partition mB of the blocks of B into partial parallel classes. That is, the containing block design is constructed in such a way that if a different resolution iiB of B is taken it may not be possible to find a resolution of the blocks of B* into parallel classes which preserve the partial parallel classes iiB of B. In 1978 the author [21] rectified this situation with the following theorem. The proof is just too technical to get into here (requiring the introduction of a considerable amount of machinery) so we content ourselves with the statement of the theorem and refer the reader to [21] for details.
Theorem 8.2 [21]. Any partial block design ( P , B ) can be embedded in a block design (S, T ) such that for any resolution T B of the blocks of B into partial parallel classes there is at least one resolution mT of the blocks of T into parallel classes which preserves the partial parallel classes mB of B. We finish up this section and set of notes by sketching a weak version of Theorem 8.1 for quadruple systems. That is, that every partial quadruple system can be embedded in a resolvable quadruple system. Actually, we can prove an analogue of Theorem 8.1 for quadruple systems: i.e., any partial quadruple system along with any partition of its blocks into partial parallel classes can be embedded in a resolvable quadruple system which preserves these partial parallel classes [19]. However, the proof is wildly tedious and so we content ourselves here with a weaker version; the proof of which is only mildly tedious. We will not bother giving obvious definitions in what follows. The following use of t h e doubling construction for quadruple systems to construct resolvable quadruple systems is used repeatedly in what follows. Let (X, x) and ( Y , y) be resolvable quadruple systems of order n where X n Y = 9 ) , F = { F , , F , , . . . , Fn-J and G = { G , , G ,, . . . , Gn-l} any two 1factorizations of X and Y, and a any permutation on {1,2,3,. . . ,n - 1). Then the quadruple system [XU V](x, y, F, G, a) is resolvable. One resolution is the following: (1) Let m = { m l , m,,. . . , m t } and .rr*={mT, m ; , . . . , mT} be any resolutions of x and y ( t = i(n - l ) ( n -2)). Then each of rIT1 U mT, T , U m:, . . . , mt Um? is a parallel class. (2) For each pair of l-factors F, and G,,where ia = j , form i n parallel classes as follows. Write
and
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{{XI, ~
2
yn-1. 3
y n } , {x39 ~
4 Y rI , Y Z } ,
...
9
{%-I,
~9
~ n - 3 ,Yn -211
is a parallel class. Evidently, the parallel classes given in (1) and (2) give a resolution of [ X U Y ] x (x, y , F, G, a ) into parallel classes. In what follows we will refer to the parallel classes given in (1) and (2) as parallel classes of type 1 and type 2 respectively.
{a, b, c, d } any block in A. Then ( X , A\{{a, b, c, d}}) can be embedded in a resolvable quadruple system (X*, A*) such that {a, b, c. d*}E A * with d E X * \ X .
Lemma 8.3. Let ( X , A ) be any resolvable quadruple system and
Proof. L e t ( Y , B ) be any resolvable quadruple system such that l X 1 = J Y Jand X fl Y = 9 and construct [ X U Y ] ( x ,y , F, G , a) exactly as in Lemma 4.1. Now let 7~ = {r,, r 2 ,... , r,}and r* = {r:, r;,. . . ,r f )be any resolutions of A and E , where {a,b, c. d } E n1and {a*, b*, c*, d * } E r T , and resolve [ X U Y](x,y, F, G, a) into parallel classes as in the remarks immediately preceding Lemma 8.3 with the following conditions on parallel classes of type 2. Place {a, b,c*, d*} and {c,d , a*, b*} in the same parallel class f l ; {a, c, b*, d*} and {b, d, a*, c*} in the same parallel class f2; and {a, d , b*, c*} and {b, c, a*, d*} in the same parallel class f 3 . Denote this resolution of [ X U Y ] ( x ,y , F, G , a ) by R. Now the only parallel classes of R that are affected in transforming [ X U Y](x,y. F, G, a ) into ( X * , A*) by unplugging (P, 0)and replacing it with the quadruple system obtained by interchanging d and d* in the blocks of Q are rl U 7 ~ 7 ,f,, f2, and f 3 . It is immediate that each of these is transformed into a parallel class of blocks of A*. Hence (R\{r, U m 7 , f l , f2, f 3 } ) U { ( r U , r T ) ’ ,f i , f;, f;} is a resolution of A * into parallel classes. where (rlU T:)‘, fl,f;, and f; are the parallel classes obtained by interchanging d and d* in the blocks of r IU r ? , fl, f2, and f 3 .
Theorem 8.4. Any partial quadruple system can be embedded in a resolvable quadruple system. Proof. Let (P, Q) be a partial quadruple system and construct a resolvable quadruple system [ X * U Y * ] ( A * ,B*, F, G , a)exactly as in the proof of Ganter’s Theorem 4.2 by replacing Lemma 4.1 with Lemma 8.3 and the appropriate replacement of “quadruple system” with “resolvable quadruple system”. Additionally we can assume that
s t ={a:, bt, c;, d}; Fit btl, [ct, 411, E-3, = { [ q ,41-[bt, ~ 3 ; Gzl= {[a?,c?1, [ b t , dll,
I? [b,,4 3, GIl = {[a:, bT1, [cT7d 3 , F2,= {[% CI
G 3 ,={[a:, d l , EbT, cT1):
Embedding theorems for Steiner systems
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{ai,bi, cT, d } and {ci,4, a:, bT} are in the same parallel class of type 2, {a,,ci, bT, d } and {bi,ci,a:, d } are in the same parallel class of type 2, and {ai,d,, by, cT} and {bi,ci, a:, d } are in the same parallel class of type 2. If we now transform [X*U Y*](A*, B*, F, G, a) into (F, exactly as in the proof of Theorem 4.2, then (P, 0)is embedded in (F, 6).A bit of reflection (but not too much) reveals that, just as in the proof of Lemma 8.3, the quadruple system (F, is resolvable.
a)
a)
We end this set of notes by stating the following theorem which is the best result to date on embedding partial quadruple systems into resolvable quadruple systems. As we have already mentioned the proof is too technical to get into here (hence not very instructive) and so we content ourselves with the statement of the theorem and refer the reader to [19] for the details of the proof.
Theorem 8.5 [19]. A n y partial quadruple system (Q,B ) along with any partition of B into partial parallel classes can always be embedded in a resolvable quadruple system which preserues these partial parallel classes. Remark. The problem of obtaining an analogue of Theorem 8.2 for quadruple systems remains an open and (so it seems to the author) quite difficult problem. References [ I ] R.C. Bose, On the construction of balanced incomplete block designs, Ann. Eugenics 9 (1939) 353-399. [21 A. Cruse, On embedding incomplete symmetric latin squares, J. Combinatorial Theory (A) 16 ( 1974) 18-27. [3] J. Doyen, Constructions of disjoint Steiner triple systems, Proc. Amer. Math. SOC.32 (1972) 409-416. [ 3 ] J. Doyen and R.M. Wilson, Embeddings of Steiner triple systems, Discrete Math. 5 (1973) 229-239. [5] B. Canter, Endliche Vervollstandigung endlicher partieller Steinerscher Systeme, Arch. Math. 22 (1971) 328-332. [ 6 ) B. Ganter, Finite partial quadruple systems can be finitely embedded, Discrete Math. 10 (1974) 397-400. [7] B. Canter, Partial pairwise balanced designs, in: Colloquio Internazionale sulle Teorie Comhinatorie (Roma, 1973). Tomo 11, 377-380. Atti dei Convegni Lincei, No. 17 (Accad. N u . Lincei, Rome, 1976). [R] J.I. Hall, J.T. Udding, On intersection of pairs of Steiner triple systems, Indag. Math. 39 (177) 87-100 ( = Proc. Koninkl. Nederl. Akad. Wetensch. Ser. A 80 (1977)). [9] M. Hall, Jr., Combinatorial Theory (Blaisdell, Waltham, MA 1967). [ 101 H.Hanani, On quadruple systems, Canad. J. Math. 12 (1960) 145-157. [ l 11 H. Hanani, The existence and construction of balanced incomplete block designs. Ann. Math. Statist. 32 (1961) 361-386. [12] H. Hanani, On balanced incomplete block designs with blocks having five elements, J. Combinatorial Theory (A) 12 (1972) 184-201. [13] H. Hanani, D.K. Ray-Chaudhuri, R.M. Wilson, On resolvable designs, Discrete Math. 3 (1972) 343-357. [14] A.J.W. Hilton, Embedding an incomplete diagonal latin square in a complete diagonal latin square, J. Combinatorial Theory (A) 15 (1973) 121-128.
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[ 151 A.J.W. Hilton, E. Mendelsohn, and L.D. Andenen, Embedding partial Steiner triple systems, (to appear). [16] T.P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. J. 2 (1847) 191-204. [17] C.C. Lindner. A partial Steiner triple system of order n can be embedded in a Steiner triple system of order 6 n + 3 , J. Combinatorial Theory (A) 18 (1975) 349-351. 1183 C.C. Lindner, Embedding block designs into resolvable block designs, Ars. Combinatoria 1 (1976) 215-219. [I91 C.C. Lindner. Every Steiner quadruple system can be embedded in a resolvable Steiner quadruple system. Ars Combinatoria 3 (1977) 75-88. r203 C.C. Lindner, Totally symmetric and semisymmetric quasigroups have the intersection preserving finite emheddability property, Period. Math. Hungar. 8 (1977) 33-39. [21] C.C. Lindner, Embedding block designs into resolvable block designs 11, AN Combinatoria 6 (1978) 4q-55. [22] C.C. Lindner. and T. Evans, Finite embedding theorems for partial designs and algebras. in: Seminaire de Mathematiques Supkrieures. Universite de Montreal (IRS Presses de I'Universitt de Montrtal. 19771 196 pp. 1231 C.C. Lindner and A. Rosa, Finite embedding theorems for partial Steiner triple systems, Discrete Math. 13 (1975) 31-39. [24] C.C. Lindner and A. Rosa, Finite embedding theorems for partial Steiner quadruple systems, Bull. Soc. Math. Belg. 27 (1975) 315-323. [ZS] R. Peltesohn. Eine Liisung der beiden Heffterschen Differenzenprobleme. Compositio Math. 6 (1939) 251-257. [26] W.B. Poucher. Finite embedding theorems for partial pairwise balanced designs, Discrete Math. 18 (1977) 291-298. [27] W.B. Poucher. A note on intersection preserving embeddings of partial (n, 4)-PBDs. Notices AMS26 (1979) A-27 (763-0.5-14). [28] R.W. Quackenhush, Near vector spaces over GF(q) and (u. q + 1. 1)-BIBDS, Linear Alg. Appl. 1 0 (1975) 25%266. I291 D.K. Rav-Chaudhuri and R.M. Wilson. Solution of Kirkman's school-girl problem, Proc. Symp. Pure Math.. I 0 (Amer. Math. Soc.. Providence, RI. 1971) 187-203. [30] D.K. Ray-Chaudhuri, and R.M. Wilson, The existence of resolvable block designs, in: A Survey of Combinatorial Theory, (North Holland, Amsterdam. 1973) Chap. 30. 361-375. [31] Th. Skolem, Some remarks o n the triple systems of Steiner, Math. Scand. 6 (1958) 273-280. [32] C.A. Treash. The completion of finite incomplete Steiner triple systems with applications to loop theory. J. Combinatorial Theory (A) 1 0 (1971) 259-265. [ 3 3 ] R.M. Wilson. Construction and uses of pairwise balanced designs, Proc. Advanced Study Inst.. Breukelen, 1974. Math. Centre Tracts, No. 55 (Math. Centrum, Amsterdam, 1974) 18-41. r341 R.M. Wilson, A n existence theory for painvise balanced designs, 111: Proof of the existence conjecture, J . Combinatorial Theory (A) 18 (1975) 71-79.
Annals of Discrete Mathematics 7 (1980) 203-21 1 @ North-Holland Publishing Company
ON THE SIZE OF PARTIAL PARALLEL CLASSES IN STEINER SYSTEMS David E. WOOLBRIGHT Special Studies Department, Columbus College, Columbus, GA 3 1907, USA
1. Introduction An S ( k , k + 1, u ) Steiner system is a pair (S, T) where S is a finite set of size u (called the order of the Steiner system) and T is a collection of (k+l)-element subsets of S (called blocks) with the property that every k-element subset of S belongs to exactly one block of T. A partial S ( k , k + 1, u ) Steiner system is a pair ( S , P) where P is a collection of ( k + 1)-element subsets of S with the property that every k-element subset of S belongs to at most one block of T. It is well known that if (S, P) is a partial S ( k , k + 1, u ) Steiner system, then IpI s(;)/(k + 1). A collection r of blocks in a S ( k , k + 1, u ) = ( S , T ) Steiner system is called a partial parallel class if the blocks in r are pairwise disjoint. If in addition the blocks of T partition S then T is called a parallel class. As a result a parallel class contains exactly u / ( k + 1) blocks. C.C. Lindner and K.T. Phelps [l] have shown that for any S ( k , k + 1, u ) Steiner system with u 3 k4+ 3 k 3 + k 2 + 1, there exists a partial parallel class containing at least ( u - k + l)/(k + 2) blocks of the system. It should be noted that ( u - k + l ) / ( k + 2 ) is equivalent to
The purpose of this paper is to improve substantially the results of Lindner and Phelps by showing that every S ( k , k + 1, u ) Steiner System with k 2 2 has a partial parallel class containing at least
(:::;:I;)(&)
- ( 2 k 3- 5 k 2 +
6 k - 1) blocks.
2. Parallel classes of Steiner systems
Let (S, T ) be an S ( k , k + 1, u ) Steiner system and let r be a collection of disjoint blocks in T with the property that if A is also a collection of disjoint blocks in T then 1 r l s l A l . Let D be the set of all elements of S which are not contained in a block of r.If we let TI= t, then ID(= u - ( k + I)?.Consider a block 203
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of the form d = A U { x } where A is any k-element subset of D. Since d is a block in a Steiner system, the choice of A uniquely determines x. Obviously x$ D, otherwise 7r U d is a collection of disjoint blocks of T with a cardinality greater than T . If we let L = { d E T ( d = A U { x } , A G D } ,then
We partition 7r into two subsets a and /3 as follows: Put c E 7r in a if c is incident with at least
2k2-k
u-(k+l)t-l k-2
blocks of L. Otherwise, we put c in p. If
blocks of L are incident with x. R = { d \ { x ) I d E L, and x E d } . It 1 . k, u - ( k + 1 ) r ) Steiner system. S ( k, k + 1 v ) Steiner system, then ~
CEQ
and
XEC,
then at most
To see this let L, = { d E L I d n c f (3) and let is easy to see that (0,R ) is a partial S ( k It is well known that if (S, Q) is a partial IQI (i)/(k + 1) and SO it follows that
Equivalently, there are at most
blocks of L which intersect x. A somewhat more surprising fact is that if c E a, then there is a unique element of c which is incident with every block of L,. To see this we recall that every block of a is incident with at least
blocks of L and so some element of c, say x, is incident with at least
blocks of L,. Suppose A U { Z } E L, with z E c, z # x. Consider the collection, G, of blocks obtained by deleting x from the blocks of L, which intersect x. Obviously (I),G) is a partial S ( k - 1 , k, u - ( k + 1)f) Steiner system. It is well known that in a (partial) S ( k , k + 1 , u ) Steiner system the number of blocks having at least one element in common with a given subset of k + 1 elements is at most
O n the size of partial parallel classes in Steiner sysrems
205
((k + l ) / k ) ( i I t ) . A direct application of this fact shows that at most
blocks of G can intersect A. Since
there must exist B E G such that B nA = $3. But then 7r U {BU{x}} U { A U { z } } / { c } is a collection of disjoint blocks in T with a cardinality greater than 1 ~ 1 . This contradicts our original assumption regarding the maximality of 1 ~ 1 .As a result A U { z } cannot exist and so every block of L, must intersect x. We have seen that at most
blocks of L, can intersect x and so we conclude that
By definition at most
2k2-k
~ - ( k +1)t- 1 k-2
blocks of L can intersect any given block of p. Since every block of L must intersect either a block of a or a block of p we conclude,
In the section above we saw that for any block c E a we could find an element x in c which was contained in every block of L,. We now assume that c,, c2, c 3 , .. . , c, are the blocks in a and that P={xI,x2, x j r . . . , x,} with xi E ci for 1s i s a and moreover each xi is incident with at least
2k2-k
(=)(
u-(k+l)t-l k-2
blocks of L. Let D' = { c ~c2, , c3, . . . , c,} U DIP and let I.' = { d E T I d = A U {x}, AED'}. Again the choice of A uniquely determines x and more importantly x$ D'. To see this suppose d = A U { x } with d c D'. If A E D then, as we have seen before, ~ $ 0 In. this case d intersects exactly one block of a. In all other cases d must intersect at least one block of a and might intersect as many as k + 1
206
D.E. Woofbrighi
blocks of a. Let d = M U N with M c D'/D and N G D. Without loss of generality we assume N f 0 and that c 1 9c , , c,. . . . . c, are the blocks of a which intersect M . We should note that r < k. Let E be the collection of blocks obtained by deleting x , from all the blocks of L , , . Trivially ( D . E ) is an S ( k - 1, k, u - ( k + 1)O Steiner system. Again we make use of t h e fact that if ( S , 7)is a (partial) S ( k , k + 1. v ) Steiner system then the number of blocks having at least one element in common with a given subset of k t 1 elements is at most ( ( k + l ) / k ) ( L .:). I n ( D ,E ) at most
blocks can intersect any given k-element subset of D. Since there are at least
blocks of L,>which intersect x, we are able to find a block B , U{X,}EL,, which does not intersect N . Now suppose q < r and we have found a collection of disjoint blocks B, U { X , } E L,,. B 2 U { x 2 L }~ c j ,.. . , B, U{X,}ELcSwhich do not intersect N. Let F be the collection of blocks obtained by deleting x q + , from the blocks of Lcq.,. Again we recall that in a (partial) S ( k , k + 1, u ) Steiner system at most ((k + l ) / k ) ( L - : ) blocks can intersect any given k reflection shows that at most
+ 1-element set.
A bit of
blocks of F can intersect N U B, U . * * U B,. Since q < r S k. q + 1 s k and so
Therefore we can find a block B , , , U { ~ , + , } E L , ~ -such , that B , U { x , } , R, U {x?}. . . . , El,, I U { x q . ,} is a collection of disjoint blocks which d o not intersect JV. This implics we can extend t h e above collection is such a way that e l = B,U{~,JEL~ , , , = B , U { ~ , } E .L.,.~. . e , = l ? , U { x , } ~ L , ,isacollection of disjoint blocks which d o not intersect N. But t h e n .rrU{e,,e,, . . . e , } U { d } l { c ,c, 2 , .. . , c,} i \ a collection of disjoint edges in (S, 7)with a cardinality greater than I T \ . We have reached a contradiction and so our assumption that d = A U { x } c D' is falsc. In other words if d = A U { x } and A E D', then x $ D'. I f L' = { d E T I d = A U { x } , A E D'}. then obviously I
Suppose x, E P, c, E SF, dnd x, E c,. How many blocks of L' can intersect x,? For any C E T let L: = { d E L'l d n c f 0 ) . If E' is the collection of blocks obtained by deleting x, from the blocks of L : ) which interscct x,, then (D'. E') is a partial
On the size of partial parallel classes in Steiner systems
207
S ( k - 1, k, u - (k + 1)t + k a ) Steiner system and so
Equivalently at most
(
u - (k
+ l)f + k a k-1
blocks of L' can intersect each xi E P. Obviously at most
blocks of L' can intersect P. If b E p, how many blocks of L' can intersect b? Suppose there are at least
[( u
- (kk+-lit+
ka
blocks of L' which intersect b. Then there is one element of b, say y, which is incident with at least
blocks of L'. If Q is the collection of blocks obtained by deleting y from every block of L; which intersects y, then (D',Q) is a partial S ( k - 1, k, u - (k + 1)t + ka) Steiner system and so
We have assumed that at least
[(u
]
+ 111 + k a ) / k +1 k-1
- (k
blocks of L' are incident with b and so there must exist a block G U {z} E Lb with z E b and z # y. If
[(u-(kk+-1;r+ka
<(A)(+ u - (k
k+l
l)f + ka - 1 k-2
D.E. Woofbrighl
208
then
(k + l ) k z u - (k + l ) f + ka - 1 k-1 k-2
)(
(
u -(k
+ l ) r + ka
k-1
(u -(k
Is(
( k+ l ) k 2
u - (k
k-1
k-2
+ l ) t + ka)(u - ( k + 1)t + ka - 1) . . . ( u - (k + 1)t + ka - k + 2) ( k - l)(k - 2 )
s
+ 1)f + ka - 1
*
*
(3)(2)
( k + l ) k 2 ( u - ( k + l ) t + k a - l ) ( u - ( k + l)r+ka-2)..,(u-(k+l)i+ka-k+2) (k - l)(k - 2 ) * * * (3)(2)
u - ( k + l)t+ k a s ( k + l)k2,
u + k a - ( k + l ) k 2 s ( k + l)t,
and finally t 3 [ u - ( k + 1)k2]/(k + 1). But then we are finished. Therefore we assume
IT[
= t 3[u -(k
+ l)k2]/(k + 1) and
The left side of the last inequality is the minimum number of blocks in Q. We recall that in a (partial) S ( k , k + 1, u ) Steiner system the number of blocks having at least one element in common with a given subset of k + 1 elements is at most ((k + l)/k)(E::). With this in mind we see that the right side of the above inequality is the maximum number of blocks in Q which can intersect G. This implies the existence of a block F U { y } with F EQ which does not intersect G U { z } . Without loss of generality let cl, c 2 , . . . , c, be the collection of blocks in a which have a non-empty intersection with F or G. Obviously s s 2 k since y. z E b and b E p. We seek to find a collection of disjoint blocks B, U{X]}EL,,, B2U {x,} E LCI.. . . , B, U {x,} E L,, which do not intersect F or G. Without loss of generality wc assume IFnDl=f#O and IGnDI=g#O. In this case s s 2 k - f - g. Consider again the collection of blocks obtained by deleting x1 from each block of Lc,. There are at least
2k2-k
(-)(
u-(k+l)r-1 k -2
such blocks. Only 0 - ( k + 1)f- 1 k-2
(&)(
of these blocks can intersect F to G and so there must exist a block B, U {x,} E L,, which does not intersect F or G. Now suppose we have found a collection of disjoint blocks B,U{X,}EL,,, B,U(X,}EL,,,. . . , B p U { x , , } ~ L c pwith l s p < s which d o not intersect F or G. Consider the collection of blocks obtained by
On the size of partial parallel classes in Steiner systems
209
Again there are at least deleting x,+, from every block in Lcp+,. v - ( k + 1)t - 1
2 k 2- k
k-2 such blocks. At most
(p + 2) k
v - ( k + 1) r - 1
(l)i
k-2
of these blocks can intersect B,U B , U f a l , g s l , we have
* *
U Bp U F U G. Since p < 2 k - f - g and
(2k- l ) k u - ( k + 1)t- 1 ( p + 2 ) k v - (k + 1)t- 1 +l>-( k - 1 k-2 k-1 k-1
)
(
and we are able to find a block Bp+,U{x,+,} such that B, U{X,}EL,,, B,U{X,}E Lc2,. . . , B,,, U { ~ , + , } E ~ is ~a , collection of disjoint blocks which do not intersect F or G. This implies we can extend the above collection in such a way that e l = B , ~ { x l } E L c , e2=B2U{x2}ELC2,.. , . , e S = B S U { x S } ~ Lisc , a collection of disjoint blocks which do not intersect F or G. But then .TrU{el, e , , . . . ,e , } u { F U { y } } U { G U { z } } l { cc,2, , .. . , c , } / { b ) is a collection of disjoint blocks in T with a cardinality greater than contradiction and so our assumption that
[(
u - ( kk+-l
1+ t
T. This
is a
ka )/kI+l
blocks of L’ can intersect b E P is false. We have shown that at most v -(k + l ) t + k a
k-1 blocks of L’ can intersect any x, E P and that at most
(
+l)t+ ka
u -(k
k-1
blocks of L’ can intersect an edge in p. These facts yield the inequality,
We now seek to relate inequalities (1) and (2). By substituting a for (r-a) for \PI in inequality (1) we see that v - (k + 1)f)
(
k
(;) ( v
s -
- ( k + 1)t)
k-1
+
(t-a)(2k2-k) ( k - 1)
(
V-(k+l)tk-2
and 1
+ l)f)(v - ( k + l ) t - k + 1) s a(u - ( k + 1 ) t )+ ( t - a ) ( 2 k 3 - k’), ( u - ( k + l ) t ) ( u - ( k + I ) ? - k + 1)-r(2k3- k 2 ) s a ( u - ( k + 1)r - 2 k 3 + (V
- (k
k2)
D.E. Woolbright
210
and finally, (U- ( k U S
+ l)r)(u - ( k + 1)t- k + 1 ) -
t(2k3- k2)
(3)
(u-(k+ l)t-2k3+k2)
By substituting u -(k
t
for
lcyl+lpl
+ 1)t+ ka
i
into inequality ( 2 ) we have 1 u - l ( k + 1)t+ k a
k-1
k
c - ( k + I ) t + k a - k + l s r , and, a s
( k + 2)t - u + k - 1 k
(4)
Combining (3) and ( 4 ) we have
( k + 2 ) t - ~ +k - l,(u k
/
( ( k+ 2 ) 1 -
t‘+
k
-
-(k+ l ) t ) ( ~ - (+ k 1)t-k+ 1)-t(2k3- k2) (u-(k+l)t-2k3+k2)
l)(u-(k
+ 1)r-2k3+ k 2 )
3k
( ~ - ( k +l ) t ) ( u - ( k + 1 ) t - k
+ 1 ) - t ( 2 k 4 - k”).
Multiplying and combining like terms we obtain the quadratic inequality below, 0>[k3
+ 3 k 2 + 4 k + 2)’
+ [ 5 k 3- k 2 - k
-
1 -(2k2+4k
+ 3)u]r
+[(k + l ) u 2 - ( 2 k 3 - l ) u + ( 2 k 4 - 3 k 3 + k2)]. ( 5 ) Obviously t is greater than or equal to the smallest root of inequality (5). In other words tz=[-b-(b2-3ac)”’]/2a where a = k 3 + 3 k 2 + 4 k + 2 , b = 5 k 3 - k 2 - k - 1 - ( 2 k 2 + 4 k + 3)u, and c = ( k + l ) u 2 - ( 2 k 3 - l ) u + ( 2 k 4 - 3 k 3 + k’). With a good bit of multiplying and combining like terms it is easy to show that b2-4ac
= U’
+ ( 8 k 6 + 4 k s - 4 k 4 -6k’
+ 6 k 2- 2 k - 2 ) ~ +(-8k7+ 13k6-10k’+11k4-5k2+2k+l).
Alternatively b2-4ac
=(u +4kh
+ 2 k 5 - 2 k 4 - 3 k 3 + 3 k 2 - k - 1)’
- ( 4 k 6 + 2 k 5 - 2 k 4 - 3 k 3 + 3 k 2 - k - 1)’
+ (-8k7 + 1 3 k 6 - 10k’+
1 l k 4 - 5 k 2+ 2 k
Since -(4k6+2ks-2k4-3k3-3k2-
k - 1)’
and (-8k7+ 13k6- 10ks+ l l k 4 - 5 k 2 + 2 k
+ 1)
are negative when k 2 2 we have b2- 4ac s (u + 4 k 6 + 2 k 5 - 2 k 4 - 3 k 3 + 3 k 2 - k - 1)2.
+ 1).
21 1
On rhe size of partial parallel classes in Sfeiner sysrems
Since t s [- b - ( b2 - 4ac)”’]/2a we have rs
-5k3
+ k 2 + k + 1 + ( 2 k 2 + 4 k+ 3 2 k 3 + 6 k 2 + 8 k+ 4 - [( u
) ~
+ 4 k6 + 2 k 5 - 2 k 4 - 3 k + 3 k 2 - k - l)’]’” 2 k 3 +6 k 2+ 8k + 4
13
( 2 k 2 + 4 k+ 2 ) ~ ( 4 k 6 + 2 k 5 - 2 k 4 + 2 k 3 + 2 k z -2k - 2 ) 2 k 3 + 6 k 2+ 8k + 4 2k3+ 6 k 2 + 8k + 4 -
IT I
= t 2 k2+
(k 2 + 2 k + I2) (L , ( 2k 3 k+l +
-
-5k2
9
+ 6 k - 1).
3. The results of Lindner and Phelps As stated in the introduction, C.C. Lindner and K.T. Phelps have shown that for any S ( k , k + 1, u ) Steiner System with u 3 k 4 + 3 k 3 + k 2 +1, there exists a partial parallel class containing at least ( u - k + l ) / ( k + 2) blocks of the system. As a corollary they were able to show that any triple system of order u 2 9 has a partial parallel class containing at least :(u - 1 ) blocks (except possibly for u = 15, 19, and 27). In the case of triple systems the results of this paper represent an improvement for all triple systems of order u s 141.
Reference [ I ] C.C. Lindner and K.T. Phelps, A note on partial parallel classes in Steiner systems, Discrete Math. 24 (1978) 109-112.
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PART V
ISOMORPHISM PROBLEMS AND ENUMERATION OF STEINER SYSTEMS
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Annals of Discrete Mathematics 7 (1980) 215-253 @ North-Holland Publishing Company
ON CYCLIC STEINER 2-DESIGNS Marlene J. COLBOURN and Rudolf A. MATHON* Department of Computer Science, Uniuersify of Toronto, Toronto, Canada
1. Preliminaries and definitions For over a century, mathematicians have been investigating the existence and construction of block designs. One of the earliest and best known references is Kirkman’s school-girl problem posed in 1850 [K3]. The more general version of arranging n objects into triples such that each pair appears exactly once was posed by both Kirkman [K2] and Steiner [S3]. As Kirkman’s paper attracted very little attention [Cll], these systems have become known as Steiner triple systems (STS). A Steiner system S(t, k, u ) is a collection of k-subsets of a u-set such that every 1-subset of the u-set appears in exactly one of the k-subsets. Over the years, a vast amount of literature has been published concerning Steiner systems, as evidenced by the thorough bibliography of Doyen and Rosa [D7]. Herein, we survey the literature relevant to cyclic Steiner S(2, k, u ) systems. We shall partition our paper into three main sections: existence, enumeration and the determination of isomorphism. In preface, we give a brief historical overview and introduce required definitions and notation. It is appropriate to mention here some related investigations which are not discussed in this survey. The first notable exclusion is the study of cyclic Steiner systems with t > 2, such as Steiner quadruple systems which are S(3,4, u ) designs. The interested reader may consult [L2, P3]. As we will see later, the study of cyclic S(2, k, u ) designs is equivalent to examining difference families with A = 1 over cyclic groups. We have omitted all research concerning difference families which either have A > 1 or are represented by using non-cyclic groups. Pointers into this literature may be found in [B16, H1, H2, B1, Cl].
1.1. A n historical overview It is elementary to establish that a necessary condition for the existence of an S(2,3, u ) is that u = 1, 3 (mod 6). Kirkman [K2] and, later, Reiss [R3] established that this condition is also sufficient. It is straightforward to demonstrate that S(2,3,7) and S(2,3,9) are unique. The first is cyclic; the second is not. The systematic investigation of cyclic STS was initiated by Netto [ N l ] in 1893. He *Research supported by NSERC Grant A8651. 215
216
M.J. Colbourn, R . A . Marhon
demonstrated the existence of two infinite families of cyclic STS, including a cyclic S(2,3, 13) and S ( 2 , 3 , 15). One of Netto’s constructions was later generalized by Heffter [H7]. Denote by N ( u , k ) ( N C ( u , k)) the number of non-isomorphic (cyclic) S(2, k, u ) systems. At the turn of the century, it was shown that N(13,3) = 2 [ Z l , D2, B171; the unique cyclic S(2,3. 13) is that of Netto. Moore [M7] showed that for u > 13, N ( u , 3) ~ 2 Cole, . Cummings and White [C6, W2] determined that N(15;3) = 80. Hall and Swift [H4] later confirmed this. Netto constructed one cyclic S(2,3, 15); in fact, NC( 15,3) = 2. The existence question for cyclic S(2,3, u ) designs remained open until 1939. In that year, Peltesohn [ P l ] constructed cyclic S(2,3, u ) systems for all u = 1 , 3 (mod 6), u f 9. Continuing interest in these existence questions has involved restricted versions of the problem. In particular, Skolem [Sl, S2] examined an “integer partitioning” problem whose solutions correspond to cyclic STS. Various extensions of Skolem’s original work have been investigated by O’Keefe [Ol ] and Rosa [R4]. Very little is known about cyclic S(2, k, u ) , k >3. Hanani [H6] demonstrated that u = 1 . 4 (mod 12) is both a necessary and sufficient condition for the existence of an S(2,4, u ) : similarly u = 1 , s (mod 20) is both necessary and sufficient for the existence of an S ( 2 , 5 , u ) . Bose [B15] constructed an infinite family of cyclic S(2, k, u ) , k = 4, 5. Heretofore, existence was resolved only for relatively small values of u. The first asymptotic determination of NC(u, 3) was pursued by Bays [B6], who restricted his attention to the case when u is prime. He demonstrated that NC(u,3) tends to infinity when u is prime. Johnsen and Storer [ J l ] have since established an exponential lower bound for NC(u,3). Similar results are not known for k > 3. Some exact values for cyclic S(2, k, u ) , determined by Bays [B6, B9], Kaufmann [ K l ] and Colbourn [C3], will be presented later. In preparation for a more detailed survey, we next present the required definitions. 1.2. Definirions A 1-design r-(u, k, A ) , is a pair (V, B)where B is a collection of k-subsets called blocks of the u-set V. such that every 1-subset of V is contained in precisely A blocks of B.A balanced incomplete block design (BIBD) is a t-design with 1 = 2. A BIBD is said to be symmetric if u = b. A Sreiner system S(t, k, u ) is a t-design with A = 1. In this paper we are especially interested in S(2, k, 21) systems. It is elementary to show that for the existence of an S(2, k, u ) design it is necessary that u - I = O (mod k - 1) and c ( u - 1 )= 0 (mod k ( k - 1)). Two such systems (V,, B,)and (V,, B,) are isornorphic if and only if there exists a bijection f : V, + V, such that b E B, if and only if f ( b )E B,. A ( u , k, A ) (cyclic) difference set D = { d , ,. . . ,d k } is a collection of k residues
On cyclic Steiner 2-designs
217
modulo u such that for any residue x+ 0 (mod u ) the congruence di - di= x (mod u ) has exactly A solution pairs (4,4 ) with 4, diE D . Every (u, k, A ) difference set generates a cyclic symmetric BIBD, whose blocks are B ( i )= { d , + i, . . . , dk + i} (mod u ) , i = 0, . . . , u - 1. The difference set is often referred to as the starter or base block of the symmetric design. A (u, k, A ) difference family is a collection of such sets D,, . . . , O n each of cardinality k such that each residue x f 0 (mod u ) has exactly h solution pairs (di, dj) with di, diED,, for some m. A difference family is said to be planar or simple if A = 1. Each (u, k, A ) difference family generates a cyclic BIBD in the same manner as before. For example, the difference family (0,1,4), (0,2,7) generates the cyclic S(2,3,13) design with V={O, 1 , . . . , 12). Using this definition, an S(2,3, 15) design cannot be represented as a difference family. However, it is possible for the design to be generated by two starter blocks modulo 15, when one includes the five blocks generated by the extra starter block (0,5,10). There are two non-isomorphic cyclic S(2,3, 15) designs generated in this manner. For our purposes, we will call a S(2, k, u ) design cyclic if the design can be generated by m starter blocks modulo u, possibly with the extra starter block (0, m‘, 2m’, . . . ,( k - 1)m‘) where b = m u + m ’ , m’
2. Existence
2.1. The existence of cyclic STS The majority of research concerning the constructions of cyclic designs has been focused on cyclic STS. In 1893, Netto [Nl] initiated the investigation of cyclic
M.J. Colbourn, R.A. Marhon
218
STS. In this early paper, Netto presented constructions for two infinite families of cyclic STS. The first is the case when u = 6 n + l and prime. Let a be a primitive element of GF(u). The construction is as follows: base blocks: multipliers:
((0, a i ,a”+’)},i = 0 , . . . , n - 1; a 2 n k ,k = 0, 1,2.
Netto’s second construction is for the case u = 3 (mod 6). A further restriction is that u = 3p where p is a prime of the form 6n + 5. Again, the base blocks are presented in terms of a, a primitive element of GF(p). base blocks: ((0, a‘,C Y ~ ” + ’ + ~ ) )i ,= 0 , . . . , 3n + 1U{(O, p, 2p)); multipliers: a k , k = 0, . . . ,6n + 4. Because t’ = 3 (mod 6), this construction includes the extra starter block (0, p, 2p). Four years after the appearance of Netto’s paper, Heffter [H7] siniplified Netto’s second construction. He investigated the case where u = 3p and p is a prime of the form 2 n + 1, pf 3. An additional requirement is that the primitive element (Y of GF(p) satisfies a = 1 (mod 3). base blocks: ((0, a i ,a n + ’ ) ) ,i = 0,. . . , n - 1 U {(0,p, 2 p ) ) ; multipliers: ak, k = 0,. . . ,2n. In the same paper, Heffter [H7]studied the construction of cyclic STS as cyclic decompositions of complete graphs. He made the observation that the construction of a cyclic S(2,3, u ) , u = 6 n + 1 , is equivalent to partitioning the set {1,. . . , 3 n } into triples such that in each triple the sum of two numbers is equal to the third or the sum of the three is equal to u. In the case u = 6 n + 3 the construction is equivalent to a partition of the set { 1, . . . , 2 n , 2n +2, . . . , 3 n + 1) with the same properties as before. These partitioning problems are known as Heffter’s first and second difference problems respectively. Complete solutions to Heffter’s difference problems were not known until Peltesohn’s paper appeared in 1939 [Pl]. We present here condensed versions of her constructions. The solutions are given in terms of the differences employed in each base block, in accordance with Heffter’s statement of the problems. u = 1 (mod 6) u = 18s + 1, s 2 2 (3r+1,4s-r+1,4~+2r+2) r = O ,..., s - 1 (3r +2,8s - r, 8s +2r+2) r = O , . . .,s - 1 ( 3 r + 3 , 6 ~ - 2 r - l , 6 s + r + 2 ) r = O , . . . ,s-2 (3s,3s + 1,6s+ 1) u = 18s + 7 , s z1
(3r+1,8s-r+3,8s+2r+4) r = O ,..., s - 1 (3r+2,6~-2r+1,6s+r+3) r = O , . .. , s - 1 (3r+3,4s-r+1,4~+2r+4) r = O ,..., s-1 (3s + 1,4s+2,7s+3)
On cyclic Sreiner 2-designs
219
u = 18s+ 13, s 3 1 (3r+2,6~-2r+3,6s+r+5) r = O , . . .,s-1 ( 3 r + 3 , 8 s - r + 5 , 8 ~ + 2 r + S ) r = O , . . ., s-1 (3r+1,4s-r+3,4s+2r+4) r = O ,..., s (3s + 2,7s + 5 8 s + 6)
u = 3 (mod 6)
*
u = 18s+3, s 1 ( 3 r + l , S s - r + 1 , 8 ~ + 2 r + 2 ) r = O ,..., s - 1 (3r+2,4s - r, 4s +2r+2) r = 0,. .. ,s - 1 (3r+3,6s-2r-l,6s+r+2) r = O , . . . , s - 1
u=18s+9,s*4 (3r+l,4s-r+3,4s+2r+4) r = O , . . . , s ( 3 r + 2 , 8 s - r + 2 , 8 ~ + 2 r + 4 ) r = 2 , . . ., s-2 ( 3 r + 3 , 6 ~ - 2 r + 1 , 6 s + r + 4 ) r = l , ..., s-2 (2,8s + 3,8s+5) (3,8s + 1,8s +4) (5,8s +2,8s + 7) , + 3,8s + 6) (3s - 1,3s + 2,6s + 1) ( 3 ~7s
1 ( 3 r + l , 4 s - r + 3 , 4 ~ + 2 r + 4 ) r = O , . . . ,s (3r+2,8s - r+6,8s +2r+8) r = 0,. . . , s (3r+3,6~-2r+3,6s+r+6) r = O , . . .,s-1. u = 18s+ 15, s *
Each of the above constructions presented for the case u = 3 (mod 6) is missing the difference iu, which corresponds to the extra starter block (0, fu,f u ) . In order to complete the solution to Heffter’s difference problems, we now present the remaining cases:
(1,2,3) (1,3,4) (2,5,6) (1,3,4) (2,6,7) (1,5,6) (2,8,9) (3,4,7) (1, 12, 13) (2,5,7) (3,8, 11) (4,6,10) (1, 11,12) (2, 17, 19) (3,20,22) (4, 10, 14) (5,8,13) (6, 18,211 (7,9,16) u = 63 (1, 15, 16) (2,27,29) (3,25,28) (4, 14, 18) (5,26,31) (6, 17,23) (7, 13,20) (8, 11, 19) (9,24,30) (10,12,22).
u=7 u = 13 u = 15 u=19 u=27 u = 45
This completes the constructions. Thus, the existence of cyclic STS is settled; there exists a cyclic S(2,3, u ) for each u = 1 , 3 (mod 6), u f 9.
2 20
M.J. C'olbourn, R . A . Mothon
2.2. Skolem's parririoning problem Skolem [Sl] posed the following restricted version of Heffter's first difference problem: partition the set { l ,. . . , 2n) into distinct pairs (a,, b,) such that b, = a, +r, r = 1 . . . . , n. If such a partitioning exists, the triples (r, a , + n , b , + n ) , r = 1. . . . , n represent a solution to Heffter's first difference problem. Skolem proved that a necessary and sufficient condition for the existence of such a partitioning is n = 0, 1 (mod 4). We give the appropriate constructions here: n =4s (4s + r - 1 , 8 s - r + 1) r = 1 , . . . , 2 s ( r , 4 s - r - 1) r = 1,. .. , s - 2 r = l , ..., s-2 (s + r + 1 , 3 s - r) (S - 1 , 3 s ) ( s , s + 1) ( 2 ~4s , - 1j (2s + 1 , 6 s )
n=4s+1 ( 4 s + r + l , 8 s - r + 3 ) r = 1,. , . , 2 s (r, 4s - r + 1) r = l , ..., s (s+r+2,3s-r+lj r = 1, . . . , s - 2 (s + 1, s + 2) (2s + 1,6s + 2) (2s+2,4s+ 1). Skolem [S 1, S2] questioned whether the above construction technique for cyclic STS could be extended to the case n = 2 , 3 (mod 4). He, in fact, conjectured that the set { 1, . . . , 2 n - 1 , 2 n + 1) could be partitioned into distinct pairs (a,, br), b, = a , + r , I = 1.. . . , n, if and only if n = 2 , 3 (mod 4). The sufficiency of this condition was proved by O'Keefe [Ol]: n=4s+2
r = 1,. . .,2s (4s + r + 3 , 8 s - r + 4 ) r = 1 , . . . , s - 1 ( 5 s + r + 2 , 7 s - r + 3 ) r = 1, . . . , s - 1 (2s + 1,6s + 2) (4s + 2 , 6 s + 3) (4s + 3,8s + 5) (7s + 3 , 7 s f 4 ) ( r , 4s - r + 2)
n=4s-l (4s + r, 8s - r - 2) r = 1, . . . , 2 s - 2 (r,4s-r-l) r = l , ...,s-2 (s + r + 1 , 3 s - r ) r = 1,. . . , s - 2 ( s - 1 , 3 ~ (s, ) s + 1 ) ( 2 ~43 , - 1) (2s + 1 , 6 ~- 1) (4s, 8s - 1). The triples (r. a, + n , b , + n ) r = 1 , . . . , n are disjoint over 11,. . . ,3n} in Skolem's constructions and over { 1 , . . . , 3 n - 1,3n + 1) in O'Keefe's constructions. Hence, thc triples (0, r, a, + n ) , r = 1 , . . . . n can be considered as the base blocks of a cyclic STS with t' = 6n + 1. The obvious question is whether such methods can be extended to the case L i r 3 (mod 6). In 1966, Rosa [R4] presented the appropriate constructions. Again, the problem can be partitioned into two subcases. Rosa demonstrated that thc set { l , .. . , n, n + 2 , . . . , 2 n + 1 } can be partitioned into pairs (a,, b,), b, = ( I , t r, r = 1.. . . , n, if and only if n = O , 3 (mod 4). The resulting triples
On cyclic Sieiner 2-designs
22 1
( r , a,+ n, b,+ n), r = 1 , . . . , n are a solution to Heffter’s second difference problem. n =4s (r, 4s - r + 1) r = 1 , . . . , s- 1 (s + r - 1,3s - r ) r = 1,. . . , s - 1 (4s+ r + 1 , 8 s - r + 1) r = 1 , . . . , s - 1 ( 5 s + r + 1 , 7 s - r + 1) r = 1,. . . ,s- 1 (2s - 1,2s) (3s, 5 s + 1) (3s + 1 , 7 +~1) ( 6 s + 1 , 8+~1) n=4~-1 (r, 4s - r) r = 1,. . . ,2s - 1 (4s+ r + 1,8s-r) r = 1,. . . ,s - 2 (5s + r, 7s - r - 1) r = 1 , . ..,s - 2 ( 2 ~6s, - 1) ( 5 ~7s , + 1) (4s+ 1 , 6 ~ (7s ) -1,7~). Similarly, the set (1,. . . , n, n + 2 , . . . , 2 n , 2n+2} can be partitioned into pairs (a,,b,), b , = a , + r , r = l , . . . , q i f a n d o n l y i f n = l , 2 ( m o d 4 ) , n # l . B e f o r e g i v i n g the general constructions we present the solutions for n = 2 and n = 5. n = 2 : ( 1 , 2 ) (4,6) n =5: (1,5) ( 2 , 7 ) (3,4) (8,lO) (9, 12) n = 4s + 1, (r, 4s - r 4-2) r = 1,.. . , 2 s (5s + r, 7s - r + 3 ) r = 1,. . . , s (4s+r+2,8s-r+3) r = l , ..., s - 2 ( 2 s + 1,6s + 2 ) ( 6 s + 1,8s + 4) (7s + 3,7s + 4)
n=4s+2 (r, 4s - r + 3)
r = 1, . . . , 2 s (4s + r +4,8s - r +4) r = 1,. . . , s - 1 ( 5 s + r+3,7s- r + 3 ) r = 1,. . . , s - 2 ( 2 s + 1 , 6 s + 3 ) ( 2 s + 2 , 6 s + 2 ) (4s+4,6s+4) (7s+3,7s+4) ( 8 s + 4,8s + 6). The triples ( r , a, + n, b, + n), r = 1, . . . , n obtained from Rosa’s constructions are disjoint over the set (1, . . . , 2 n , 2n + 2 , . . . ,3n + 1) in the case n = 0 , 3 (mod 4) and are disjoint over the set (1,. . . , 2 n , 2 n + 2 , . . . , 3 n , 3 n + 2 } when n = 1, 2 (mod 4). As before, these triples correspond to a cyclic S ( 2 , 3 , u ) , u = 6 n + 3, n f 1; the design is generated by the base blocks (0, r, a, + n), r = 1,. . . , n, in conjunction with the additional base block (0, $u, $u). Hence, the partitioning problems discussed herein give us another technique for constructing cyclic STS. 2.3. Existence of cyclic S(2, k , u ) , k > 3 Very little is known about cyclic S(2, k , u ) systems when k > 3. Bose [BlS] has constructed two infinite families of such systems. The first is for k = 4; u is prime
M.I. Colbourn. R.A. Marhon
222
and of the form 12n + 1. Let a be a primitive element of GF(u),which satisfies f f 4 n - 1 = a q for some odd q. base blocks:
(0, a", a4nc2', aRn+2' ), i = O , . . . , n - 1 .
Bose's second construction is very similar to the first; k = 5 and u is prime of the form 20n + 1. The primitive element a of GF(u) must satisfy a4"+ 1 = aq where q is odd. base blocks:
(a2t,
a4n+21
ff12n+21
a8n+21 3
,
i = 0 , . .. , n - I .
We now present without proof two new constructions.
Theorem 2.1. Let u = 4p. where p is a prime of the form 1 2 n + 1. Let a be a primitive element of GF(p) satisfying a = 3 (mod 4). The 4n + 1 blocks
a4'+9, i = 0,. . . , 3 n - 1,
( 0 , a'', (0,
1, f f 4 " + 4 t + 1
,
),
i = O ...., n-1,
( 0 ,p, 2P, 3P)
are the base blocks of a cyclic S(2.4, u ) system.
Theorem 2.2. Ler u = 5p, where p is a prime of the form 4n + 1. Let a be a primitive elentent of GF(p) satisfying a = 4 (mod 5 ) and such that (a' + 1)= a d ( a C - 1 ) for some odd integers c, d . The n + 1 blocks ( 0 ,a 2 1 , f f z I + c f f 2 n + 2 , f f Z n f Z , i < ), i = O , . . . , n - 1 , (0, p, 2p, 3p, 4p)
are the base blocks of a cyclic S(2.5, u ) system. These are the only general constructions known for k > 3. Although individual cases have been settled it is not known whether, for a fixed k >3, a cyclic S(2, k, u ) design exists for each admissible order of u. Cyclic S(2, k , u ) systems, k = 3 , 4 , 5 for several orders of u will be presented in the next section.
3. Enumeration Although not all of the existence questions are settled, research is well underway on the related enumeration problems. As in the case of existence, STS have received far more attention than Steiner 2-designs with larger k. STS are, in fact. the only S ( 2 , k, u ) designs for which asymptotic bounds are known. We will present these results first, then turn our attention to individual cases where exact values are known.
On cyclic Steiner 2-designs
223
Doyen [D4, D5] was the first to demonstrate that N(u, 3) tends to infinity as u does. This result was later improved upon by Doyen and Valette [D6] and Wilson
W31. The analogous problem for cyclic STS was first investigated for certain subcases. In the previous section we discussed the existence of Skolem sequences. Recall Skolem’s problem: partition the integers { 1, . . . , 2 n } into pairs (a, b,), b, = a, + r, for r = 1, . . . , n. Skolem [Sl] proved that such a partitioning is possible if and only if n =O, 1 (mod 4).From these partitions, one can obtain cyclic STS. Hence an investigation of the number of distinct partitions for a given n, could lead to a lower bound for NC(u,3). These partitions have been examined by Davies [Dl], Markov [Ml], Hanani [H5], and Alekseev [All. The latter author established that the number of distinct (unordered) partitions does, in fact, grow exponentially. Distinct partitions do not necessarily lead to non-isomorphic designs. However, this result does lend evidence to the belief that NC(u, 3) will also exhibit exponential growth. The other subcase extensively investigated is that of prime orders. Throughout the 1920’s and 1930’s, Bays wrote prolifically regarding STS of prime order [B3-B12]. In one of his early papers [B6], Bays indicated that N a u , 3 ] * [2”-’/n], when u = 6 n + l is prime. We will discuss other aspects of Bays’ research later. Despite Bays’ early success, little progress was made on the general case until Johnsen and Storer [Jl] established that for u = 1 , 3 (mod 6), u # 9. 3), 2~/6-1/2-(log,(r:+l))~
~ c (. ~ ,
This bound follows from their investigation of CIP neofields (cyclic neofields which possess the inverse property). Thus, Johnsen and Storer have successfully demonstrated that NC(u,3) does tend to infinity (exponentially) as u does. Unfortunately, no such results are known for higher k. Bays pioneered the determination of exact values for NC(u, 3). In [B3-B8], he first discussed the development of an heuristic procedure for finding many non-isomorphic cyclic STS. In this series of papers, he obtained exact values for NC(u, 3), u = 1 (mod 6), u ~ 3 1 A. colleague of his, Kaufmann [Kl], obtained similar values for NC(u,3), u = 3 (mod 6), u S 3 3 . We will present these values later. In a second series of papers [B9-B12], Bays extended and formalized this heuristic approach to enumerate cyclic STS of prime order. He claimed that, at least in principle, his technique could be used to mechanically determine NC( u, 3) for prime values of u. Subsequent investigations have refuted this claim [C3], since for u = 43 Bays’ technique counts 9380 designs. NC(43,3) is, in fact, 9508. Nevertheless, it appears that Bays method can be employed to obtain excellent lower bounds in specific cases. Recently, a computer investigation of NC(u, k) was undertaken by the authors. The first phase includes the production of a catalogue of all inequivalent designs
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for given I: and k. This exhaustive list is obtained via an orderly algorithm [R 1, C 2 ] ; backtracking techniques are employed to generate cyclic designs in lexicographic order. The lexicographically smallest representation of a particular design is obtained by examining all multiplier automorphisms. If the design under consideration is already in minimum form (lexicographically smallest), it is added to the catalogue. Otherwise, it is discarded, as we have already encountered a lexicographically smaller representation. The orderly procedure can easily be modified to obtain the order of the multiplier group; this is just the number of times the minimum form is encountered. After a complete catalogue of inequivalent designs is generated, nonisomorphism must be verified. Verification of non-isomorphism and canonization of these designs will be discussed extensively in the next section. We now present the known values of NC(v, k ) , k s 5 . (We know of no enumeration results regarding k > 5 with the exception of certain uniqueness theorems of projective planes and certain non-existence proofs for cyclic S ( 2 , 6 , v ) designs [M6]).
7 9 13
1s 19 21 25 27 31 33 37 39 43 45
1
1
2 4 7 12 8 80 84 820 798
c'
NC(u.4)
13 16 25 28 37 40 49 52 61 64
2 10 224 206 18132 12048
1
v
21 25 41 45 61
65
NC(u.5) 1
1
10 2
9508
11616
The appendix contains a catalogue of all cyclic S(2, 3, v ) v ~ 3 3 S(2,4, , u) v s 52, and S ( 2 . 5 , v ) v s 6 5 .
4. Determination of isomorphism
Unlike the research of Bays and Kaufmann, exhaustive enumerations of designs are generally performed with the aid of a computer. The crucial step in such a generation is the elimination of isomorphic (duplicate) designs. Various general techniques, such as orderly algorithms [ R l , C 2 ] , and the exploitation of design symmetry [B13, G 2 , G 3 , M4] can be used to dramatically reduce the magnitude of this task. Nevertheless, a large number of designs will typically remain indistinguished. Hence, a method for deciding design isomorphism is required. In this
On cyclic Steiner 2-designs
225
section, we outline the current knowledge concerning the computational complexity of this problem. 4.1. Complexity of isomorphism testing
The graph isomorphism problem is widely believed to be difficult [R2], although it is not known to be NP-complete [Gl]. Recent investigations have demonstrated that many problems are polynomial-time equivalent to the graph isomorphism problem [B 141. Such problems are termed isomorphism complete. Demonstrating that a problem is isomorphism complete is usually considered to be strong evidence that there is no polynomial time algorithm to solve it [C9]. In 1968, Corneil [CS] observed that, in practice, graphs derived from block designs form one of the hardest subcases of the graph isomorphism problem. Other researchers have since corroborated this viewpoint [R2, M2]. Only recently, block design isomorphism [C4] and isomorphism of 4-class association schemes [Fl] were shown to be isomorphism complete. Hence, we are unlikely to find an efficient (polynomial-time) algorithm for block design isomorphism. Consequently, we are motivated to search for better algorithms for specific subcases. Using a result of Tarjan, Miller [M5] showed that quasigroup isomorphism can be decided in subexponential time; the standard representation of an STS as a Steiner quasigroup yields a subexponential algorithm for deciding isomorphism in this case. A detailed description of this method and an extension to S(t, t + 1, u ) designs are given in [C3]. In this paper we are especially interested in cyclic designs; certain improvements can be made in this case. Recall that it is unknown whether equivalence and isomorphism are synonymous. There is an elementary polynomial time algorithm for deciding equivalence of two difference families. Hence, if all inequivalent designs are non-isomorphic, there would be a polynomial-time algorithm for deciding isomorphism of difference families. Prior to examining this question in more detail, we consider related problems concerning graphs. A transitive graph with a prime number of vertices is known to be a circulant [Tl]. The only automorphisms of a prime circulant correspond to multipliers [ E l , D3]. The automorphisms of a circulant with pq vertices ( p , q distinct primes) are also multipliers [A2]. However, circulants with p2 vertices (p>3) may have non-multiplier automorphisms. In fact, a stronger statement can be made. There exist isomorphic inequivalent circulants [El]. Let us return to cyclic designs. There exist cyclic STS which have non-multiplier automorphisms, e.g. the 2-transitive S(2,3,31). Despite the existence of these non-multiplier automorphisms no pair of inequivalent isomorphic designs is known. The Bays-Lambossy theorem [B9, L l ] which we prove next, guarantees that such a pair does not exist on a prime order.
Theorem 4.1 [B9, Part 111. Given 2 isomorphic cyclic structures on a prime number of elements, there exists a multiplier isomorphism transforming one to the other.
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M.J. Colbourn, R.A. Marhon
Let C represent the cyclic group on u elements; S, the symmetric group. The metacyclic group M is the group of substitutions of the form x-ax + b (mod u ) , (a, u ) = 1, 0s b s ( u - 1). The normalizer of C in S is the group N , ( C ) = {s 1 s-'Cs = C, s E S}. It is easy to verify that N,(C) = M. Consider the following 4 systems of conjugates: AG={s-'Gs I SES},
BG = {n-'Gn 1 n E N s ( C ) } , DG={~-'CSIS-'C~
FG = { g - ' c g I g E G}.
Lemma 4.2. Suppose C < G < S and C < s - ' Gs for all s E S, then DG = FG implies A, = BG. = {C, C,, . . . , C,}.Consider a conjugate s-'Gs, and its subgroups s-'Cs, s-'Cls,. . . , s-'C,s. From the assumption C < s-'Gs, C must be one of the subgroups s-'C,s given above. If C = s-'Cs,we can conclude s E Ns( C). Now consider the case C = s-'C,s for some i, 1G i s r. From the assumption DG = F G , we know that C = s-'(g-'Cg)s for some g E G. Hence, C = (gs)-'C(gs) which implies gs E N,(C). We know s 'Gs = s-'g-'Ggs for each g E G. Hence, s-'Gs = K ' G n for some n.
Proof. Form the set DG; let DG SES
Lemma 4.3 (Sylow, see [H2, Chapter 41). Let G be a group of order pkm, (p, m ) = 1. All subgroups of order p k form a single system of conjugates.
Proof of Theorem 4.1. Consider 2 isomorphic cyclic structures on p elements with automorphism groups G, and G, respectively. C C GI, i = 1 , 2 . If G , = C, N,(G) = N,(C) = M . Since lGll I IS(,we know lG,I = pm, (p, m ) = 1. From Lemma 4.3, all p-Sylow subgroups of G, form a single system of conjugates. Hence, Dct = FG,. Hence by Lemma 4.2, AG, = Bol. Clearly, G,is a conjugate of GI in S and, therefore, is a conjugate of G , in N , ( C ) = M. We conclude that there is at least one transformation from the metacyclic group N,(C) which transforms G, to G,.This establishes the result. It is important to note that Theorem 4.1 is a statement about cyclic hypergraphs, a broad class of structures incoiporating both circulants and cyclic designs. Using Theorem 4.1 we observe that there is an O(u2)algorithm for deciding isomorphism of cyclic designs with a prime number of elements. In determining this complexity, we assume that the algorithm is given a cyclic representation of each design; the complexity of recognizing cyclic designs is unknown to the present authors. In practice, this does not create any difficulty since one usually deals with a difference family representation of the design.
On cyclic Sfeiner 2-designs
227
Until the question of equivalence and isomorphism is resolved, we must resort to isomorphism decision procedures for general block designs. 4.2. lnuarianrs The lack of a polynomial time algorithm for block design isomorphism compels us to search for techniques which reduce the magnitude of this problem. In particular, given a list of designs we require a method of partitioning the list into classes such that two isomorphic designs are in the same class. We prefer an heuristic method which results in small classes. Consider the following example. For our list of designs, we compute the automorphism group. We then partition the designs so that two designs are in the same class if and only if they have the same number of automorphisms. Since isomorphic designs have the same number of automorphisms, they must be in the same class. Design properties such as this are isomorphism invariants. We view an invariant as a function I for which l ( D l ) =I(DJ if D, and D2 are isomorphic. When I ( D l ) = l ( D 2if) and only if D, = D2, the invariant I is complete. There is no known efficiently computable complete invariant. To maintain efficiency in resolving isomorphism we must, at present, resort to incomplete invariants. In choosing such an invariant we wish it to reduce the magnitude of the problem as much as possible. With this in mind, Petrenyuk and Petrenyuk [P2] propose that a measure of the invariants effectiveness be its sensitivity-the ratio of the number of classes it distinguishes to the number of non-isomorphic designs under consideration. A complete invariant has sensitivity one. In the remainder of this section we will consider invariants with respect to ease of computation and sensitivity. We first examine invariants for block designs in general and then investigate invariants specifically for Steiner systems. In most cases we will present numerical evidence in support of our statements concerning the sensitivity of the invariant. Some of these results are presented in [C3]. 4.2.1. Invariants of block designs
One of the earliest invariants employed was the order of the automorphism group. This invariant is insensitive as demonstrated by the following statistics for cyclic STS. U
sensitivity: size of largest class:
15
19
21
25
27
31
33
1
0.75
0.86
0.08
0.12
0.05
0.05
1
2
2
12
8
63
78
A second difficulty is that no polynomial time algorithm is known for computing the order of the automorphism group. In fact, there is evidence that computing
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228
the order of the automorphism group is equivalent to deciding isomorphism; in the related case of graphs, computing the order of the automorphism group is isomorphism complete [Bl, M3]. In specific instances a small number of designs can be partitioned by hand using this invariant. For example, the 2 cyclic S(2.3, 15) designs are distinguished by the orders of their automorphism group. Nevertheless, the complexity of the computation in general makes it unsuitable for large numbers of designs. Another means of distinguishing designs is by examining the number and type of subdesigns. Moore [M7] used this invariant to demonstrate the existence of at least two non-isomorphic STS, u > 13. This invariant is also insensitive. r sensitivity: size of largest class:
15
19
21
25
27
31
33
1
0.25
0.57
0.08
0.12
0.04
0.01
1
4
3
12
8
78
84
From a complexity standpoint this is a poor invariant; there is no known polynomial time algorithm for deciding whether one design is a subdesign of another. The corresponding problem for graphs is NP-complete [C7]. A further difficulty with subdesigns is that one must examine relatively large “pieces” of the design. A generalization called fragment analysis [G2] circumvents this difficulty. A specific instance of fragment analysis for STS will be discussed later. Gibbons employed another class of invariants, observing that invariants of the block intersection graph are in fact invariants of the design. Given a design D, we can define a series of block intersection graphs Gi,1 = 0 , . . . , k, defined as follows: The vertices of Gi are the blocks of D. Two vertices are adjacent if and only if the corresponding blocks contain exactly i elements in common. One effective invariant is to count the number of cliques of size c in G,; this will be referred to as (c, i)-clique analysis. Gibbons employed (4,O)-clique analysis with great success, indicating that the invariant is quite sensitive. For cyclic STS, c s 2 7 . (4, ())-clique analysis is a complete invariant. The complexity of this invariant is also appealing; an O(b4) algorithm for computing this invariant is immediate. When the design is transitive, we need only consider the number of cliques containing a particular element. Hence, an O(rb3)algorithm results. The practical difficulty, however, makes the invariant difficult to use. In a (4,O)-clique analysis of cyclic STS, we found the 2-transitive S(2,3, 15) contains 394 cliques, whereas one of the S ( 2 , 3 , 2 1 ) designs contains 24646 cliques. The number of cliques for relatively small values of u is enormous. Thus although the growth is
On cyclic Steiner 2-designs
229
polynomial the computation is extremely expensive. Furthermore, it appears that in order to maintain high sensitivity, the size of cliques being examined must increase as a function of u. If this is indeed the case, the computation is extremely difficult from a complexity standpoint-it is, in fact, a special case of a #Pcomplete problem [Vl, Gl]. 4.2.2. Invariants for S(f, t f 1, u ) In 1913, White [Wl] introduced a method of distinguishing the two S(2,3, 13) designs. Given a STS D, consider a triad (x, y, z) which is not in D. (x, y, z) is transformed by replacing each pair (x, y), (x, z), (y, z) by the single element with which it appears in D. Another triad results. For example, let D contain the three triples (1,2,4), (1,3,5), (2,3,6); the triad (1,2,3) will be transformed into (4, 5 , 6 ) . If one continuously repeats this operation, one of two things must occur. Either a triad of D is encountered or a previous triad is again reached. For simplicity, White refers to triads of D as one term cycles. Hence, every triad not in D initiates a frain of triples which terminates in a periodic cycle. Examining these trains allowed White to differentiate the two S(2,3, 13) designs. Clearly, two isomorphic designs will have the same train structure. However, the converse is not known to be true. In order to determine a design's train structure, each triad must be examined once. Hence, an O(u3)algorithm results. White proposed this invariant simply for STS; however, the obvious extension allows one to investigate trains for S(t, f + 1, u ) designs in O(u'+')time. Another invariant introduced to distinguish STS is the graph of interlacing. (We will use the term cycle structure.) Cole [C5], Cummings [ClO], Hall and Swift [H4] employed this invariant to distinguish STS of small orders; we describe it here in a more general setting. For a given S ( f, f + 1, u ) Steiner system D = ( V, B ) , consider any set A c V such that IAl= t - 1. For convenience let A ={x,, . . . , x * - ~ } .We define a graph GA to be G(VA, EA) where
VA = V- A
and E A
= {(a,b )
I a, b E VA,
. . . ,x,- 1, a, b ) E B}.
(XI,
This graph is a 1-factor. Given D, consider two sets of elements A = { x l , . . . , x , - ~ } and C = {x,, . . . , x ~ - x~l }, . We define GA = (V,, EA)and G, = (V,, E,) as above. We now define the union of two such graphs GA U Gc to be G(V', E', L ) where v ' = VA n v C - { x
I (XI,.
..
9
XI, X ) E
B}
and
I
E' = {(a, b ) a, b E V', (a, b ) E EA or (a, b ) E Ec}
and L is a mapping of edges to labels. L(a, b) = A if (a, b ) E E A . Because every t-tuple must appear exactly once in D, each element x in V' appears once in a block with the set A and once with the set B. Hence, GA UG, is regular of degree 2; it is therefore a union of cycles.
M.J. Colbourn. R.A. Mathon
230
A compact notation for this graph is just the list of cycle lengths in ascending order. This is called the cycle list for the pair of ( t - 1)-sets A and C.Consider the cycle lists for every pair of ( t - 1)-sets, which have t-2 elements in common. This collection of lists. when ordered lexicographically, is called the cycle structure. For cyclic STS one only has to consider the cycle lists for the pairs (0, i), 1=sic ;(u - 1). For SQS, this invariant has been used by Phelps [P3] to distinguish the 29 S(3,4,20)designs. For STS, Petrenyuk and Petrenyuk [P2] define an equivalent invariant, T tables. They pose t h e question of determining the sensitivity of this invariant. In the case of cyclic STS, cycle structure is quite sensitive, although not complete. , sensitivity is approximately 0.9. For L' ~ 4 5 its
L'
sensitivity: F i x
15
19
21
25
27
31
33
37
39
43
45
I
1
0.714
0.917
I
0.913
0.940
0.911
0.915
0.901
0.911
1
1
3
2
1
4
5
of the
largest class:
50
36
464
S17
There is an elementary O(u3)algorithm for computing this invariant for STS. (O(u2)for cyclic STS). Its high sensitivity guarantees the existence of many classes containing a single design. It has the added attraction that even for classes containing more than one design, a subexponential isomorphism algorithm based on cycle structure can be employed to differentiate the designs [C3]. In order to appreciate the effectiveness of cycle structure, it is worthwhile to examine the diversity of the encountered cycle structures. Let us, therefore, consider potential cycle lists. Clearly, every cycle has even length 5 4. Hence, the number of potential cycle lists is exactly the number of partitions of ( u - 3 ) into even integers >2. In our studies we found that almost all potential cycle lists actually appear in a cyclic STS:
Li
31
33
31
39
43
4s
85.3
85.4
89.4
89.9
92.7
89.7
% of potential cycle lists which
appear in cyclic STS
It appears from this table that the asymptotic growth rate of the number of cycle lists agrees with that of the partitions. This rapid growth, together with the many ways of combining cycle lists to form cycle structures accounts, in some sense, for the sensitivity of cycle structure. It would be interesting to establish that asymptotically all potential cycle lists are realized in an STS. This would provide strong evidence that cycle structure remains sensitive for higher orders.
On cyclic Steiner 2-designs
23 1
4.2.3. Invariants for Steiner systems In this section we consider arbitrary Steiner S(t, k, u ) systems. The invariants from the previous section are applicable only when k - t = 1. However, when this is not the case we can still define the graph GA, / A [ =t - 1. GA is a collection of disjoint ( k - t + 1)-cliques. We may again define the labelled graph GA U G,, as before. Any invariant of this graph is an invariant of the pair of ( 1 - 1)-sets A and C. For a given invariant Z, let Z(A, C) denote the value of Z on GA U GC An invariant of the design is the multiset {Z(A, C) 1 ( A l = t - 1, ICJ=t - 1, ( An CI= t-2, A c V, C C v>. One can see that cycle structure is an invariant of this form. Let us consider a specific graph GAU Gc When k - t > 1 there does not appear to be a convenient notation €or the graph (i.e. one similar to cycle lists). We will, therefore, examine weaker invariants of the graph. Let X be the ( k - t + 1)-clique common to both GA and G,. Initially, we arbitrarily order the ( k - t f 1)-cliques of GA- X. Now each ( k - t + 1)-clique K in G, - X can be represented as a ( k - t + 1)-set S ( K ) ; if u belongs to the ith clique of GA - X and u E K, i E S ( K ) . Observe that for u, w E K, vf w, v and w belong to different cliques in GA - X . Hence, S(K) is a ( k - t + 1)-set. For a given i, consider the k - t + 1 sets S ( K , ) ,. . . , S(Kk-,+,)which contain i. From this collection form T ( K j )= S ( K , ) - { i } . Now T(K,), . . . , T(Kk-r+,)form the edges of a ( k - 1)-uniform hypergraph, which we will denote Hi and call an ouerlap graph. Any invariant of the collection {Hi} is an invariant of GA UG,. Each overlap graph Hihas the same number of edges, so this invariant would result in no discrimination. However, they may have a different number of vertices. With this in mind, we define the overlap list of GA U G,, OL(A, C), to be the multiset {I V(Hi)l}.The overlap list is clearly invariant under isomorphism. The overlap structure of a design is the multiset {OL(A, C ) 1 JAl=r - 1, ICI= r - 1, IA n C J =t-2, A c V, C c
v}.
A seemingly more powerful invariant can be defined by enumerating all ( k - t)-uniform hypergraphs with (k - t + 1) edges and arbitrarily ordering them 1 through m. For such a hypergraph H denote by # ( H ) its index in this list. The The typed typed overlap list of GA U Gc,TOL(A, C), is the multiset {#(Hi)}. ouerlap structure is the obvious analogue of overlap structure. We will illustrate the above definitions by an example. Consider an S(2,4,40). Let A = ( 0 ) and C = { 1). GA is composed of disjoint triangles, similarly G,. One numbers all triangles of GA, excluding the triangle containing 1, one through r - 1. A given triangle K of G,, excluding the triangle containing 0, is then represented as a triple S ( K ) of numbers from one through r - 1. We have r - 1 such triples. Consider l G i G ( r - 1). Let S ( K , ) , S ( K , ) , S ( K 3 ) be the triples containing i. Deleting i from each, we obtain three 2-sets T ( K , ) , T ( K , ) , T ( K 3 ) . These form the edges of the 2-uniform overlap graph Hi. In determining the
232
M.1. Colbourn. R.A. Mothon
overlap list OL(A, C ) we examine only the number of vertices in each Hi, 1s i s ( r - 1). In computing the typed overlap list TOL(A, C), we first enumerate the eight multigraphs with three edges, number them 1 through 8, and represent each overlap-graph Hiby its index in the list. With respect to ease of computation there is an efficient algorithm for computing this invariant. Furthermore, in our investigations of cyclic designs we found the invariant to be extremely sensitive. In fact, for cyclic S(2,4,u ) , u s 6 4 , the overlap structure distinguishes all designs. Considering typed overlap lists, it is interesting to note that in the cyclic S(2,4, u ) designs investigated, all possible , structure is again 3-edge multigraphs appear. For cyclic S(2,5, u ) , u ~ 6 5overlap complete. To our knowledge, this invariant has not previously appeared in the literature.
4.3. Concluding remarks
Our investigation of cyclic S(2, k, u ) designs has indicated that overlap structure is a very sensitive invariant. Applying such an invariant to a catalogue of designs aids in distinguishing them. Isomorphism testing may, nevertheless, be required if the invariant is not complete. When u is prime, however, this is not the case, since the Bays-Lambossy theorem guarantees that we need only decide equivalence (via an elementary polynomial time algorithm). Any non-trivial generalization of this theorem could potentially obviate much of this isomorphism testing. Alspach and Parsons’ theorem suggests an appropriate first step: the resolution of the case when u is a product of distinct primes.
5. Open problems
We recall here open problems mentioned in the text 5.1. Existence
(1) Find other infinite families of cyclic S(2, k, u) designs for k 2 4. (2) Show that the necessary conditions for a cyclic S(2,4, u ) design are sufficient for all u b u ’ , u’ = 37. In other words, show that these necessary conditions are sufficient for all but a finite number of orders. (3) Find the smallest u‘ as above for any fixed k >4.
S.2. Enumeration (1) Does N C ( u , k ) grow exponentially for every fixed k?
(2) Can Skolem’s partitioning problems be generalized to yield cyclic S(2, k , u ) designs, k = 4,5?
On cyclic Steiner 2-designs
233
5.3. Isomorphism (1) Do there exist isomorphic inequivalent cyclic S(r, k, u ) designs? (2) Generalize the Bays-Lambossy theorem to non-prime orders. (3) Can isomorphism of arbitrary Steiner systems be decided in subexponential time? (4) Given an arbitrary design, what is the complexity of deciding whether it is cyclic? ( 5 ) What is the asymptotic sensitivity of various invariants such as trains, cycle structure and overlap structure?
6. References [A l l V.E. Alekseev, Skolem method of constructing cyclic Steiner triple systems, Math. Notes 2 (1967) 571-576. [A21 B. Alspach and T.D. Parsons, Isomorphism of circulant graphs and digraphs, Discrete Math. 25 (1979) 97-108. [Bl ] L. Babai, On the isomorphism problem, unpublished. [B2] L.D. Baumert, Cyclic Difference Sets, Lecture Notes in Mathematics No. 182 (Springer-Verlag, Berlin, 1971). [B3] S. Bays, Sur les systkmes cycliques de triples de Steiner, C.R. Acad. Sci. Paris, SCrie A 165 (1917) 543-545. [B4] S. Bays, Sur les systemes cycliques de triples de Steiner, C.R. Acad. Sci. Paris, Strie A 171 (1920) 1363-1365. [BS] S. Bays, Sur les systtrnes cycliques de triples de Steiner, C.R. Acad. Sci. Paris, Strie A 175 (1922) 936-939. [B6] S. Bays, Recherche des systtmes cycliques de triples de Steiner difftrents pour N premier (ou puissance de nombre premier) de la forme 6n + 1, J. Math. Pures Appl. (9) 2 (1923) 73-98. [R7] S. Bays, Sur les systbmes cycliques de triples de Steiner, Ann. Sci. Ecole Norm. Sup. (3) 40 (1923) 55-96. [B8] S. Rays, Sur les systemes cycliques de triples de Steiner difftrents pour N premier (ou puissance de nombre premier) de la forme 6n + 1, Ann. Fac. Sci. Univ. Toulouse (3) 17 (1925) 23-61. [B9] S. Bays, Sur les systtmes cycliques de triples de Steiner diffkrents pour N premier (ou puissance de nombre premier) de la forme 6n + 1, I, Comment. Math. Helv. 2 (1930) 294-305. 11-111, Comment Math. Helv. 3 (1931) 22-41. IV-V, Comment Math. Helv. 3 (1931) 122-147. VI, Comment Math. Helv. 3 (1931) 307-325. [R10] S. Bays, Sur les systbmes cycliques de triples de Steiner differents pour N premier de la formes 6n + 1, Comment Math. Helv. 4 (1932) 183-194. [Bl I] S. Bays, Sur le nombre de systtmes cycliques de triples differents pour chaque classe W . Actes SOC.Helv. Sci. Nat. 116 (1935) 275-276. [Bl2] S. Bays, Sur les systtmes de caracteristiques appartenant a d = 3, Actes SOC.Helv. Sci. Nat. 116 (1935) 276-277. [B13] T. Beyer and A. Proskurowski, Symmetries in the graph coding problem, Proceedings of the Annual Northwest ACM/CIPS Conference (1976). [B 141 K.S. Booth and C.J. Colbourn, Problems polynomially equivalent to graph isomorphism, Technical Report CS-77/04, Dept. of Computer Science, University of Waterloo (1979). [B15] R.C. Bose, On t h e construction of balanced incomplete block designs, Ann. Eugenics 9 (1939) 353-399. [B16] R.H. Bruck, Difference sets in a finite group, Trans. Amer. Math. SOC.78 (1955) 464481. [B17] G. Brunel. Sur les deux systtmes de triads de treize elements, J. Math. Pures. Appl. 7 (1901) 305-330.
234
M.J. Colbourn, R.A. Marhon
[Cl ] P. Camion. Difference sets in elementary abelian groups SCminaire de Mathtmatiques Suprieures, Montreal (1979). [C2] C.J. Colbourn and R.C. Read, Orderly algorithms for graph generation, Int. J. Comp. Math., to appear. [C3] M.J. Colbourn, An analysis technique for Steiner triple systems, Proceedings of Tenth Southeastern Conference on Combinatorin, Graph Theory and Computing (1979). [C4] M.J. Colbourn and C.J. Colbourn, Concerning the complexity of deciding isomorphism of block designs, submitted tor publication. [CS] F.N. Cole, The triad systems of thirteen letters, Trans. Amer. Math. SOC. 14 (1913) 1-5. [C6] F.N. Cole, L.D. Cummings and H.S. White, The complete enumeration of triad systems in 1.5 elements, Proc. Nat. Acad. Sci. U S A . 3 (1917) 197-199. [C7] S.A. Cook, The complexity of theorem-proving procedures, Proceedings of the Third ACM Symposium on the Theory of Computing (1971) 51-58. [C8] D.G. Corneil, Graph isomorphism, Technical Report 18. Dept. of Computer Science, University of Toronto (1968). [C9] D.G. Corneil, Recent results on the graph isomorphism problem, Proceedings of the Eighth Manitoba Conference on Numerical Mathematics and Computing (1978). [ClO] L.D. Cummings, On a method of comparison for triple-systems, Trans. Amer. Math. SOC.1.5 (1914) 311-327. [Cl I ] L.D. Cummings. An undervalued Kirkman paper, Bull. Amer. Math. SOC.24 (1918) 336-339. [Dl] R.O. Davies, On Langford's problem, Math. Gaz. 4 3 (1959) 253-255. ID21 V. DePasquale. Sue sistemi ternari di 13 elementi, Rend. R. ist Lombard0 Sci. e lett. 32 (1x99) I! 13-221. [D3] D.Z. Djokovic, Isomorphism problem for a special class of graphs, Acta Math. Acad. Sci. Hung. 21 (1970) 267-270. [D4] J. Doyen, On the number of non-isomorphic Steiner systems S(2. rn, n ) , in: Combinatorial Structures and their Applications, Proceedings of the Calgary International Conference (1969) 63-64. [DS] J. Doyen. Sur la croissance du nombre de systemes triples de Steiner non isomorphes, J. Combinatorial Theory 8 (1970) 42-41. ID61 J. Doyen and G. Valette, On the number of non-isomorphic Steiner triple systems, Math. 2. 120 (1971) 178-192. [D7] J. Doyen and A. Rosa, An extended bibliography and survey of Steiner systems, Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (1977) 297-361. [ E l ] B. Elspas and J . Turncr. Graphs with circulant adjacency matrices, J. Combinatorial Theory 9 (1970) 297-307. [FI] M. Fontet. private communications (1979). [G l ] M.R. Carey and D.S. Johnson, Computers and Intractability; a Guide to the Theory of NP-Completeness (Freeman, San Francisco, 197Y). [GZ] P.B. Gibbons, Conputing techniques for the construction and analysis of block designs, Technical Report 92, Dept. of Computer Science, University of Toronto (1976). [G3] P.B. Gibbons, R.A. Mathon and D.G. Corneil, Computing techniques for the construction and analysis of block designs, Utilitas Math. 11 (1977) 161-192. [H l ] M. Hall, Jr., A survey of difference sets, Proc. Amer. Math. SOC.7 (1956) 97.5-986. [HZ] M. Hall. Jr.. The Theory of Groups (Macmillan, New York, 1959). [H3] M. Hall, Jr., Combinatorial Theory (Wiley. New York, 1967). [H4] M. Hall, Jr. and J.D. Swift, Determination of Steiner triple systems of order 1.5. Math. Tables Aids Comput. 52 (1955) 146-1.52. [ H S ] H. Hanani. A note on Steiner triple systems, Math. Scand. 8 (1960) 154-156. [H6] H. Hanani. The existence and construction of balanced incomplete block designs. Ann. Math. Statist. 32 (1961) 361-386. [H7] L. Heffter. Ueber Tripelsysteme. Math. Ann. 49 (1897) 101-1 12. [J I] E.C.Johnsen and T. Storer, Combinatorial Structures in Loops: IV Steiner triple systems in neofields. Math. Z. 138 (1974) 1-14. [ K I ] P.B. Kaufmann, Studien iiber zyklische Dreiersysteme der Form N = 6n +3, InauguralDissertation der Math.-Natur. Fakultat der Universitat Frieburg in der Schweiz, Sarnen (1926).
On cyclic Steiner 2-designs
235
[K2] T.P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. J. 2 (1847) 191-204. [K3] T.P. Kirkman, query, Lady’s and Gentleman’s Diary 48 (1850). [Ll] P. Lambossy, Sur une manibre de diffkrencier les fonctions cycliques d’une forme donnte, Comment. Math. Helv. 3 (1931) 69-102. [L2] C.C. Lindner and A. Rosa, Steiner quadruple systems-a survey, Discrete Math. 21 (1978) 147-1 8 1. [Ml] A.A. Markov, A combinatorial problem, Problemy Kibernetiki 15 (1965). [M2] R.A. Mathon, Sample graphs for isomorphism testing, Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory and Computing (1978) 499-5 17. [M3] R.A. Mathon, A note on the graph isomorphism counting problem, Inf. Proc. Lett. 8 (1979) 131-132. [M4] B.D. McKay, Backtrack programming and the graph isomorphism problem, M.Sc. Thesis, Math. Dept., University of Melbourne (1976). [M5] G.L. Miller, On the nlogn isomorphism technique, Proceedings of the Tenth ACM Symposium on the Theory of Computing (1978) 51-58. [M6] W.H. Mills, The construction of balanced incomplete block designs, Proceedings of the Tenth Southeastern Conference on Combinatorin, Graph Theory and Computing (1979). [M7] E.H. Moore, Concerning triple systems, Math. Ann. 43 (1893) 271-285. [Nl] E. Netto, Zur Theorie der Tripelsysteme, Math. Ann. 42 (1893) 143-152. [Ol] E.S. O’Keefe, Verification of a conjecture of Th. Skolem, Math. Scand. 9 (1961) 80-82. [Pl] R. Peltesohn, Eine Losiing der beiden Heffterschen Differenzenprobleme, Compositio Math. 6 (1939) 251-257. [P2] L.P. Petrenyuk and A.Y. Petrenyuk, An enumeration method for non-isomorphic combinatorial designs, Annals of Discrete Math. 7, this volume. [P3] K.T. Phelps, On cyclic Steiner systems S(3,4,20), Annals of Discrete Math. 7. [Rl] R.C. Read, Every one a winner, Annals of Discrete Math. 2 (1978) 107-120. [R2] R.C. Read and D.G. Corneil, The graph isomorphism disease, J. Graph Theory 1 (1977) 339-363. [R3] M. Reiss, Uber eine Steinersche combinatorische Aufgabe, welche im 45sten Bande dieses Journals, Seite 181, gestellt worden ist, J. reine angew. Math. 56 (1859) 326344. [R4] A. Rosa, Poznamka o cyklickych Steinerovych systemoch trojic, Math. Fyz. Cas. 16 (1966) 285-290. [Sl] Th. Skolem, On certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957) 57-68. [S2] Th. Skolem, Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958) 273-280. [S3] J. Steiner, Combinatorische Aufgabe, J. reine angew. Math. 45 (1853) 181-182. [Tl] J. Turner, Point-symmetric graphs with a prime number of points, J. Combinatorial Theory 3 (1967) 136145. [Vl] L.G. Valiant, The complexity of computing the permanent, Theor. Comp. Sci. 8 (1979) 189-20 1. [Wl] H.S. White, Triple-systems as transformations and their paths among triads, Trans. Amer. Math. SOC.14 (1913) 6-13. [W2] H.S. White, F.N. Cole and L.D. Cummings, Complete classification of the triad systems on fifteen elements, Memoirs Nat. Acad. Sci. U.S.A. 14, 2nd memoir (1919) 1-89. [W3] R.M. Wilson, Nonisomorphic Steiner triple systems, Math. Z. 135 (1974) 303-313. [Zl] K. Zulauf, Uber Tripelsysteme von 13 Elementen, Dissertation Giessen, Wintersche Buchdruckerei Darmstadt (1897).
Summary of the groups of cyclic S(2.3, v )
MULT
AUT
N(r)
c
7
1
168
21
13
1
39
39
15
1
1
60 20 160
60 60
1 2 1
19 57 171
19 57 171
1
1
21 47 I26 504 882 1oox
21 42 126 63 63 126
25
12
25
25
27
8
27
27
31
63 15 1 1
31 93 465 9999360
31 93 465 155
33
7x 3 2 1
33 66 165 330
33 66 330
37
777 42 I
37 111 333
37 111 333
39
730 4 55 2 4 2 1
39 78 117 156 234 468 3042
39 78 117 1S6 234 468 117
43
9377 129 1 1
43 129 30 1 903
43 129 30 1 903
11616
45
45
19
21
1
2 1 1
4s
165
Legend. N ( u ) is the number of cyclic S(2.3, u) with the specified AUT and MULT. AUT is the order of the automorphism group. MULT is the order of the multiplier group. 236
C y c l l c steiner S ( 2 , 3 , v )
v =
%?sLgns,
v = 7 t.?.roiigL 7 ?
7
1 1 1
0
1
3
I
1
I
1
I
I
II
11 II
v = ?1 1
I1
2 11 3 11 4 5 6 7
11 11 11 11
0
1 7 1
3
1
1
n
0 0 0 0
1 5 1 1 5 1
0 3
1
5
1
0
1 1
9 7
1 1
0
3
7
0
0
4
1 2 1
3
171
?
1 3 1
2 3 ?
1 0 1
-
D 0
5’11 5 ’ 3
I
3 ’ 1 7 1 5 1
1 3 1
0 3 3
5 1
3
U ’ 7 I
r I
3
1 1 ’ 7 1
7
‘ 1
I 1 I I I I I
II
50u
II II
21
II II I I II
120s
63
126 1 1
1
11 11 1I 126 1 1
1
12h “47 63 i? 4? !.?b
126
11
a
I I I I
7
II II II II
71 1 1
1
Legend. AUT is the order of the automorphism group, MULT is the order of the multiplier group, D, is the number of S ( 2 . 3 , u ) subdesigns.
I t
Y n
0
i;’
v = :> 11 I1
II
II II I I II
II II 11 II II
II II II 11 II I1 II
II v = 71
1 1
11 II 11
\ I II II
!I !I 1 1
I I II
m
n
O n cyclic Sreiner 2-designs
I-
239
240
m r(
3
m
.r
n
a
k3 .4
M.J. Colbourn, R.A. Marhon
“ I
c
c
s
0 0 0 C 0 0
it c1
aa
s a j
as
J
fa
Y . J 3
r)
On cyclic Steiner 2-designs
3 3
a
zi
r\i N
r ‘ r :r: c.” ri
PIP
24 1
r J ri
R
a
a
a
oooocooccoocooooooooooooooooooocYooo~O
zi
0 0
oooooooooo~oooooooooooooooooooooooooc
:Y
00
IJ N
I I I I
I I I I II II I I 11 I1 II II
I1
If II
I I II I I 11
I I I I I1 II 11 1 1 II 1 1 II
I I
ir
5
p
5 P
?
3
$ I
On cyclic Sreiner 2-designs
Cyclic s t e i n e r S ( 2 . 4 . v )
243
v = 17 t h r o u g h 5 2
Designs,
v = 13 111
0
1
0 0
3
3 2 4 1 1 3 2 4 1
0 0
4 9 1 5 1 4 3 6 3 7 1
0 0 0 0
1 1 1 1 1
0
2 2 2 2 2 ? 2
3
9 1
I
I
I
v = 37 111 2 1 1
3
0
7 1 7 2 5 1 ' 1 7 2 5 1
v = 40 111 2 1 1 3 1 1 4 1 1
5 1 1 6 1 1
711 8 1 1
9 1 1 1011
0
0 0 0 0 3
1 1 1 1
1 1 1 1 7 a 1 5 1 5 1 5 1 5 1
1 1 1 1 1
3 3 3 3 3
1
4 4 4 4 4
3 7 3 T 3 3 7 7 7 7
1 ) 1 I 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0
1 1 1 I
7 7 7 7 8 S 9
2 2 2 2 2 3 1
4 4 4 U 5 5 5
1 1 1 I 1 1 1
O
I
S
I
0 0 0 0
2 2 7 3 3 1 2 2 7 3 3 1
0 0 0 0 0 0
4 1 8 2 ' 3 1 U I R 2 7 ( 0 4 7 8 2 9 1 0 4 1 8 2 3 1 0 4 2 0 3 0 ) 0 4 2 0 3 9 1 0 4 2 0 3 0 1 0 4 2 0 3 0 1 0 4 2 3 3 7 1 0 4 2 3 3 3 1 0 4 2 3 3 3 1 0 4 2 3 3 3 1 0 4 2 4 3 5 1 0 ' 4 2 4 3 5 1 0 4 2 4 3 5 1 0 a 2 4 3 5 1 ! I 4 1 6 3 5 I 0 4 1 6 3 5 1 0 4 1 6 3 5 1 0 4 1 6 3 5 I 0 4 1 8 3 7 1 0 4 1 8 3 7 ) 0 U 1 8 3 7 1 0 4 1 8 3 7 1 0 4 2 0 2 7 1 0 4 2 0 2 7 1 0 4 2 0 2 7 1 0 4 2 0 2 7 1 0 4 2 6 3 3 1
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
z
6 6 5 5 5 6 3 3 3 3
1 4 2 5 1 2 1 3 2 1 1 1 1 9 1 2 6 3 4 1 1 1 1 9 1 1 4 2 5 1 1 4 2 7 1 3 1 2 1 1 1 4 2 2 1 212:,1
b 6 6 5 6 6 0
? l 3 3 / ,71331 2 2 3 4 1 2 2 7 U 1 1 7 3 1 1 1 7 3 1 1 2 4 7 8 1
I I 1 1 I 1 I I
I
I
v = 49 111 2 1 1 3 1 1 4 1 1
5 6 7 0
1 1 1 1
1 1 1 1
9 1 1 1011 1111 1211 1311 1411 1511 1611
1711 1811 1911 '011
2111 2211 2311 2411 2511 2611 2711 2811 2911
0 0 0 C 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
8 1 9 1
8 1
R I 8 1 R I
1
3
1 1 1 1 1 1 1 1 1 1 1 1 1
3 3 3 3 3 3 7 3 3 3 3 3 3
1
3
O
1 1 1 1 1
3 3 3 3 3 3 3 1 3
9 1
f
1 1 1
8 1 8 1 9 1
S l 8 1 R l 8 1
3 8 8 3
1 1 1 1
" 1 9 1 1
9 1
3 1
q I " J
9 1 ' I 9 1 9 1
0 0
3 1 9 3 2 1 0 3 2 h 3 3 1 0 9 1 9 3 7 1 0 3 2 6 3 3 1 0 9 2 1 3 6 1 0 0 2 2 3 7 1 0 9 2 1 3 6 1 6 z u 3 9 1 0 3 2 2 3 7 1 6 1 7 3 1 1 0 9 2 1 3 6 1 6 1 7 3 1 1 0 9 2 2 3 7 1 6 2 4 3 8 1 0 9 2 1 3 0 1 6 2 4 3 R 1 0 3 2 2 3 7 1 6 2 1 3 3 1 0 9 1 9 3 2 1 6 2 1 3 3 1 0 9 2 6 3 q 1 6 2 2 3 4 1 0 9 1 9 3 2 1 6 , 2 2 3 4 1 0 9 2 6 3 9 1 5 2 2 2 9 1 0 1 0 2 1 3 6 1 5 7 2 2 9 1 0 1 3 2 3 3 S 1 5 2 5 3 2 1 0 1 0 2 1 3 6 1 5 2 5 3 2 1 0 1 0 2 3 3 8 I 5 2 2 2 9 1 0 1 0 2 1 3 6 1 5 2 2 2 9 1 0 1 0 2 3 3 8 1 5 2 5 3 2 1 O l O 2 1 3 6 ( 5 2 5 3 2 1 0 1 0 2 3 3 8 1 5 1 7 3 5 1 0 1 0 2 1 3 4 1 5 1 7 3 5 1 0 1 0 2 5 3 8 1 5 1 ? 3 7 1 0 1 0 2 1 3 4 1 5 1 9 3 7 1 0 1 0 2 5 3 8 1 5 1 7 3 5 1 O f 0 2 1 3 4 1 0
244
M.J.Colbourn, R.A. Mothon
O n cyclic Steiner 2-designs
9 9 3 9
0 1 2 3
1 1 1 1
1 1 1 1
3 4 1 )
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Annals of Discrete vathematics 7 (1980) 255-264 @ North-Holland Publishing Company
NON-ISOMORPHIC REVERSE STEINER TRIPLE SYSTEMS OF ORDER 19 R.H.F. DENNISTON Deparrmenf of Mafhematics, Uniuersiry of Leicesfer, Leicesfer LEI 7RH, England
1. Introduction It has been known since 1917 that, up to isomorphism, there are just 80 Steiner triple systems of order 15. The next admissible order for such systems is 19; but the corresponding problem is too big to be feasible, and it may not be without interest if a restricted form of it can be solved. One severe restriction is to require a system to have what we may call a “reversal”, namely an involutory automorphism with only one fixed point. Triple systems with reversals have in fact been given the name of reverse Steiner triple systems, and have recently been the subject of various papers, beginning with [ 13 and [4].The present paper establishes that the maximum number of non-isomorphic reverse systems of order 19 is 184.
2. Uniqueness of reversal
Theorem 1. If a Steiner triple system of order u has two different reversals, then u is divisible by 3 . Proof. Let r and s be the fixed points of different reversals p and a. Then points x and y correspond in p if and only if rxy is a line; so s must be distinct from r, since otherwise a would be the same as p. Let R be the group of automorphisms generated by p and a: one orbit under R consists of r, s, and the point r collinear with r and s. Let x be any other point. Then a p x and papx are in line with r ; using a, we find that px and a p p x are in line with t ; using p, we find that x is in line with s and papapx, and therefore coincides with a p p p x . So we see that, in the orbit to which x belongs, there are just six points. That is, the group R partitions the set of u points into orbits, each of cardinality three or six, and t h e Theorem is proved. We may observe that apa is a reversal with t as fixed point, and that R is naturally isomorphic with the group of permutations of { r , s, t}. An affine space of order three and any number of dimensions has, of course, as many reversals as it has points. I have found it easy to construct, on 33 points, a system with at least 255
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R.H.F. Dennisron
three reversals; and I conjecture that such a system exists whenever u is congruent to 3 or 9 modulo 24.
3. Projections and coverings Let S be a reverse Steiner triple system of order 19. We suppose that S has its points so named that its reversal is the permutation
Then we may construct a design D on nine varieties 1 , 2 , . . . , 9 , by taking in turn each pair of corresponding lines (not through 0) of S, and simply dropping the primes. If, for instance, 1’2‘3’, 1”2”3“, 1’2”4‘, l”2’4” are lines of S, then 123 and 124 are blocks of D. We naturally call D “the projection” of S ; we see at once that, given any two of the nine varieties, we can find just two distinct blocks of D to which they belong. So the projection D is a balanced incomplete block design without repeated blocks, and its parameters are given by v=9,
b=24,
k=3, r=8,
h=2.
The situation would be complicated if a system could have more than one reversal, and so more than one projection: but Theorem 1 reassures us that this will never happen with a system of order 19. We can therefore see that the projections of two isomorphic reverse systems will be isomorphic ( 9 , 2 4 , 3 , 8 , 2 ) designs. So,if we are to solve the problem of non-isomorphic reverse systems, we should begin with the problem of non-isomorphic (9,24,3,8,2)-designs without repeated blocks. After that, we should consider the problem of setting up “a covering” for a given design D - namely, a reverse system S which has D as its projection. We shall see that there is a design which has no covering, but that other designs can be covered in 213 or 214 ways. The appropriate point of view will be that the varieties of D are numbered 1 , 2 , . . . , 9 in a fixed way, and that the labelling 0, l‘, . . . .9“ of the points of the covering S is required to be consistent with this; also that different arrangements into triads of the symbols 0. l‘, . . . , 9 ” are to be regarded as “distinct” coverings. Suppose, then, that S is a covering of 0,and that we interchange the two symbols 1’ and 1” wherever they occur in S. This will give a covering of D, distinct from S according to the convention just made, but isomorphic with S as a Steiner system. We might say that we have “re-primed” the variety 1 ; and we could go o n to re-prime other varieties of D. However, the result of re-priming all nine varieties would be a reverse system which, by definition, coincided exactly with S, On the other hand, we shall certainly get a covering distinct from S if we re-prime any proper subset of ( 1 , . . . ,9}. To prove this, we have only to consider
Non-isomorphic rewrse Steiner triple systems of order 19
251
the effect on the four lines 1’2’3’, 1”2”3”, 1’24‘, l”2‘4“ of any re-priming that includes 1 but not 2-and to remember that any two varieties of D will be covered by four lines in essentially this way. It follows that, by taking S as equivalent to any covering obtained from S by re-priming, we partition the set of all coverings of D into equivalence classes (“prime classes”, shall we say), each of cardinality 2’ exactly. Two coverings of D, if they are in the same prime class, are isomorphic as reverse Steiner systems. Now let p be an automorphism of the given design D. Then we can transform any covering S into an isomorphic reverse Steiner system, p*S say, by making the obvious permutation of the points: if, for instance, 1 4 2 in p, then 1 ’ 4 2 ‘ and 1”+2” in p * , while 0 is fixed. Moreover, p* preceded by the re-priming of 1, and p* followed by the re-priming of 2, will in this instance have the same effect: we can assert that p will permute the prime classes in a well-defined way. If p happens to fix the prime class of a given covering S, we can get an automorphism of S if we follow p* with a suitable re-priming: and conversely. We shall see that, when a design D has coverings, a non-trivial automorphism of D never fixes all t h e prime classes, though it may fix a quarter of them. The conclusion is that the automorphism group of a design D will act as a group of permutations on the set of prime classes of coverings of D. If the prime class of a covering S is in the same orbit as that of a covering S*, then the reverse Steiner systems S and S* are isomorphic. Conversely, we remarked above that isomorphic systems have isomorphic projections: if we think concretely of two isomorphic coverings of the same design, we see that their prime classes are in the same orbit. Or, if we are given two isomorphic reverse systems that are specified in some other way, we can label their points so that they will have the same design as projection - and then, over that design, they will be in the same prime class (or, at least, in two prime classes in the same orbit). We can now see how the investigation should go. First, we exhibit a maximal set of non-isomorphic (9,24,3,8,2)-designs without repeated blocks, each design having its automorphism group determined. Then, for each of these designs, we find the number of prime classes, and sort these into orbits under the automorphism group (by looking for cases where an automorphism can fix a class). The total number of orbits will be the required number of non-isomorphic reverse Steiner systems.
4. A set of designs A solution to the problem of non-isomorphic designs, obtained by means of a computer, has been published by Gibbons [2]. I had independently solved the problem by hand, and have now checked that my solution agrees with Gibbons’. I will write out the designs as I found them, with Gibbons’ Roman numbers in brackets.
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D,: 123 124 134 156 159 167 178 189 234 256 259 267 (I) 278 289 357 358 368 369 379 457 458 468 469 479 We do not need the automorphism group of D , (which is of order 80). D,: 123 129 134 145 156 167 178 189 234 246 257 258 (VII) 267 289 357 358 368 369 379 459 468 478 479 569 Group of order 2 generated by (12) (39) (48) (5) (67).
D,:
123 129 134 145 156 167 178 189 234 246 258 259 (IX) 267 278 357 358 368 369 379 457 468 479 489 569 Group is trivial. D,:
123 129 134 145 156 167 178 189 235 246 247 257 (HI) 268 289 348 359 367 369 378 458 469 479 568 579 Group of order 2 generated by (18) (2) (36)(45) (7) (9).
D,: 123 129 134 145 156 167 178 189 235 246 247 257 (IV) 268 289 348 358 367 369 379 459 469 478 568 579 Group is trivial. Dh: 123 129 134 145 156 167 178 189 234 246 257 259 (V) 268 278 357 358 368 369 379 458 467 479 489 569 Group is trivial. D,: 123 129 134 145 156 167 178 189 235 246 247 257 (VI) 268 289 348 358 367 369 379 459 468 479 569 578 Group of order 2 generated by (14) (28) (3) (5) (69) (7).
D,: 124 127 136 139 146 157 158 189 235 238 245 268 (X) 269 279 347 349 356 378 458 467 489 569 579 678 Group of order 6 generated by (123) (456) (789) and (14) (26) (35) (7) (89). D,: 126 127 134 139 145 158 168 179 237 (XI) 256 289 348 356 359 367 467 469 478 Group of order 8 generated by (1 2345678) (9).
238 245 249 578 579 689
Ill,,: 124 127 136 138 145 159 169 178 235 238 249 256 (XII) 267 289 346 349 357 379 457 468 478 568 589 679 Group of order 6 generated by (123456) (789). D l l : 123 126 139 147 148 156 157 189 234 247 258 259 (VIII) 268 279 345 358 367 369 378 456 469 489 579 678 Group of order 6 generated by ( I 23456) (789).
Non-isomorphic reverse Steiner triple systems of order 19
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D12: 123 125 134 148 157 168 169 179 236 247 249 259 (XIII) 268 278 349 357 358 367 389 456 458 467 569 789 Group of order 18 generated by (123) (456) (789) and (174) (235689). D13: 126 129 137 139 147 148 156 158 237 238 246 248 (11) 257 259 346 349 356 358 457 459 678 679 689 789 Group of order 8 generated by (1234) (5) (67) (89) and (1) (24) (3) (5) (68) (79). 5. Construction of coverings
The problem of finding coverings, for a given design D, will be easier to handle if we set up some kind of arithmetic for it. Let us think again of the example at the beginning of Section 3, where 123 and 124 are blocks of D. We may express the assertion that each of the pairs { l’,2’) and { l”,2”) is collinear with a point that projects onto 3, while {l’,2”) and 11”. 2’) are collinear with points that project onto 4, by writing [l, 2; 31 = + 1,
[ 1,2; 41 = -1.
The symbol [p, q ; r ] is defined only when pqr is a block of the given design D. The following equations are satisfied when the symbols in them are defined:
[P, 4 ; rl = [q, p; rl. [ p , q ; r l . [p, q ; s I = -1
-
(1) ( r f s),
[P, 4 ; r l . [r, p; ql Eq, 7; PI= + I .
(2) (3)
In fact, we can see the truth of (3) by imagining that we have tossed three coins, and want to take one away so as to leave a head and a tail; there will be either two ways of doing this, or none. Equations (l),(2), (3) are not only necessary, but sufficient: if we have written down symbols [p, q; r ] , three for each block of D, that satisfy these equations, we have effectively set up a covering of D. A topologist might visualise the blocks of D as two-dimensional simplexes of a pseudomanifold, and might arrange a covering by setting up a one-dimensional cochain. However, a dual interpretation can be made in terms of graph theory. Let us consider the blocks of D as the vertices of a graph, two vertices being adjacent when the blocks have a pair of points in common [3]. Eleven of our designs will then have connected graphs; one exception is DI3,where the four vertices 789, 689, 679, 678 make up one component, and the other twenty make up a second. The graph of D, has three components: one consists of ten vertices 156, 167, 178, 189, 195, 256, 267, 278, 289, 295; another has ten vertices, and the third has four. The choice of a covering €or D will now correspond to the standard process of converting the graph into a directed graph (will “orientation” serve as a name for
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this process?). Taking once more the example symbolized by the equations [ 1.2; 31 = + I and [ 1.2; 41 = - 1, we may say that the edge which represents {1,2} has been oriented, the head end being at the vertex which represents the block 123, and the tail end at 124. Equation (3) now tells us that, of the three edges through any given vertex, either all have their head ends there, or one has a head end and two have tail ends. A simple counting argument then shows that, in any component of such a directed graph, exactly a quarter of the vertices have the former property (of being at the head ends of all their edges). But this means that a graph. if ten of its vertices make up a component, cannot possibly be oriented according to t h e rule. We conclude that there is no reverse system that has D , as its projection. As it turns out. each of the other twelve designs has some coverings, and accordingly has a graph for which there is, shall we say, at least one “admissible orientation”. This fact, which we shall establish in the next section, does not seem easy to infer from the graph theory. However, it does seem obvious what happens when G. the graph of a design D, has been provided with two admissible orientations A , and AZ. Let C be the set of edges of G which we have to reverse in changing from A , to A,. Then at any vertex, since the number of head ends is odd for A , , and likewise for A,. there must be an even number of edges that belong to C.That is, C will be a cycle of G. Conversely, given any cycle C of G, and one admissible orientation A , , we can change from A , to another admissible orientation by reversing all the edges of C. It follows that a given graph with first Betri number v, i f it has any admissible orientations at all, will have exactly 2” of them To find v, we subtract 24, the number of vertices of any one of our graphs, from 36, the number of edges, and add the number of components: so v is 13 for each of the designs from D, to D,, inclusive, and 14 for DI3.According to section 3. the respective numbers of prime classes are 2’ and 2‘, for the coverings of these designs.
6. Construction of coverings: an effective procedure It remains to be seen how the symbols [ p , q ; r ] can be used to handle these large sets of coverings. Let us take the design D, by way of example. We may choose to say that a covering of D, is “standardised” when the lines through 7’ are 7’7”0, 7’1’2”. 7’2’9”, 7‘9’5”. 7’5’1’’, 7’4’6, 7‘6’8”, 7’8‘3’’, 7’3‘4“. It is in fact easy to see that, in any prime class of coverings of D,, there is one and only one standardised covering. For such a covering of D,, we see that [ 1 , 2 ; 71 = - 1; and it follows, by equation (2) of section 5 , that [ l , 2; 41 = + I . Likewise, [2,9;61 = + 1,
[ 5 , 9 ;61 = + 1,
[ 1 , 5 ; 81 = + 1,
Non-isomorphic reverse Steiner triple systems of order 19
26 1
Let us now give names to five (suitably chosen) of the other symbols; it turns out that a convenient set is a = [2,8; 61,
fi = [8, 9; 41,
6 = [3,9; 41,
= [l,9; 81.
y = [2,5; 31,
Then we should be able to find all the other symbols in terms of these. First, we use Eq. (2) again:
[2,8;3]=-a,
[8,9; 1]=-/3,
[3,9; 1]=-6,
[1,9;5]=-~.
[2,5;4]=-y,
Now, using Eq. (3) as well, we deduce from the equations [3,8; 21 = +1 and [2,8; 3]= -a that [2,3; 81 = -a, whence again [2,3; 51 = a. Likewise, we find successively that
[2,6; 81 = a, [ 5 , 6; 91 = a, P - 6 ; 51 = Y, [4,8;9]=-p6, [ 5 , 8 ; 11= B E , [2,4; 5]=-y&, [ 1,6; 41 = - Y ~ E ,
[2,6; 9]= - a ; [5,6;3]=-a, [3,6; 1]=-7, [4,8;5]=@; [5,8;4]=--P~; [2,4; 1 ] = ~ 6 ~ [ 1,6; 31 = y& ;
[6,9; 2]= -a, [3,5;2]=ay, [4,9;3]=6, [1,8;9]=-/3~, [4,5; 8]=-&, ;[1,4;2]=~6~, [ 1,3; 61 = - 6 ~ ,
[6,9; 5]= a ; [3,5;6]=-ay; [4,9; 8]= -6; [1,8;5]=@, [4,5; 2 ] = 6 ~ ; [1,4;6]=-yb; [ 1,3; 91 = SE.
In the course of this arithmetic, we have ensured that condition (3) is satisfied for all but one of the 24 blocks of D,. We now observe that
[3,9; 1]=-6,
[1,9;3]=-~,
[1,3;9]=6~;
so condition (3) is satisfied for the last block, and there is no doubt that the system exists. Each of the symbols a,p, y, S, E can take the values +1 and -1 independently. So we have effectively constructed 2’ coverings of D8,all of them standardised as above. No two of these are in the same prime class, and we can obtain from them, by re-priming, the complete set of 213 “distinct” coverings of D,. It turns out that the same algorithm works for each of the designs D,, . . . , DI3,except that D,, with its higher Betti number needs six Greek letters.
7. Effect of an automorphism
Our next task, according to the programme at the end of Section 3, is to determine how the automorphism group of a design acts on the set of prime classes of its coverings. In particular, we should consider how often a non-trivial automorphism will leave a prime class fixed.
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By way of example. suppose that C , a prime class of coverings of D8,is fixed by the automorphism (123) (456) (789). In C there is one system, S say, which is "standardised" as in Section 5. According to Section 3, we are concerned with the "obvious" transformation 1'+2', 1"+2", 2'+3', . . . , determined by our automorphism. This will take S to a system, 7-say, which by hypothesis is also in the class C. Let us choose one of the two re-primings that take T back to S ; and let us write nq= -1 if the re-priming interchanges q' and q", but nq= +1 if it does not. Now the standardised system S will certainly have 7'9'5" as one of its lines; this will correspond to a line 8'7'6 in T, and the re-priming will take this to one or the other of two lines 7'6'8" and 7"6"8' that belong to S. So definitely n,r6 = - 1. In terms of the symbols a. 0, y , 6, E that belong to S, and of its numbers [p, q ; r], we can work out necessary and sufficient conditions for the re-primed system to be a standardised one: [ 9 . 3 ; 11 = -6, [9, 1: 81 = E , [9,2: 6]= + I , [9.6; 51 = a, [9.4; 31 = 6, [ 9 , 5 ;71-1, [9,7:2]=-1, [9.8; 41 = p,
[7,1: 2]= + l ; [7,2;9]=+1; [7,3; 41 = + 1; [7 , 4 ;6] = + 1; [7,5; 1 ] = + 1 ; [7,6; 81 = + 1; [7,8; 3]= + I ; [7,9; 51 = + 1;
These equations determine the set of numbers 7rq (to within multiplication of the whole set through by -1). But we are assuming, not merely that the re-primed system is standardised, but that it actually coincides with S. So we need five more conditions:
This is as far as we need to go, because the standardised covering of D, is uniquely determined by its symbols a, p, y, 6, €.'The only restrictions on S arc thc last five, which in fact reduce to two independent conditions a = ps = E .
The conclusion is that (123) (456) (789) fixes just a quarter of t h e prime classes of coverings of D, (namely those whose standardised representatives satisfy these two conditions). The same arithmetic, applied to the involution (14) (26) (35) (7) (89) of D,, leads to an impossible condition a = -a. So this automorphism does not fix any
Non-isomorphic reverse Steiner triple systems of order 29
263
prime class. It turns out, likewise, that no prime class is fixed by any involution of any standard design (nor, consequently, by any automorphism of period 4 or 8).
8. Counting the systems Accordingly, we see in the case of D9that the automorphism (12345678) (9) must permute the 32 prime classes in four orbits of length 8; and this means that, up to automorphism, D9is the projection of only four reverse systems. Likewise, each of the designs D,, D4, D,, having an automorphism group of order two, is the projection of 16 systems. For D13, there are 64 prime classes, permuted by the eight automorphisms in eight orbits of length 8; so D,, is the projection of eight reverse systems. Going back to D,, we see that its automorphism group has two generators, one of order three fixing eight of the 32 prime classes, and one of order two fixing none. It follows that there are four orbits of length 2 and four of length 6. And therefore a maximal set of non-isomorphic coverings of D, will consist of eight reverse systems, four having automorphism groups of order 6 (which turn out to be cyclic), and four of order 2. The designs D,, and D,,, each of which has an automorphism group of order 6, give results analogous to those for D,. Each of these has eight nonisomorphic coverings, of which four have cyclic automorphism groups of order 6 . The design D,,, with a group of order 18, has two prime classes which are fixed by all its automorphisms of period 3 (but interchanged by all its involutions). Then there are six classes that are fixed by (123) (456) (789), but the automorphism group is transitive on this set of six; and likewise for (147) (258) (369). Finally, there is a set of 18 prime classes on which the group is sharply transitive. The conclusion is that D,, has four non-isomorphic coverings, with automorphism groups of orders 18, 6, 6, 2. These groups turn out to be Abelian, even though the group for D,, is not. To summarise, we have 32 non-isomorphic coverings for each of the designs that have trivial automorphism groups, namely D,, D,, and D,. We have 16 for each of D,, D4, D,; eight for each of D,, Dlo,D I 1DI3; , four for each of D, and D,,: but none for D,.
Theorem 3. Up to isomorphism, there are just 184 reuerse Steiner triple systems of order 19. One of these has an automorphism group of order 18 (Abelian, not cyclic), and fourteen have (cyclic) groups of order 6. For each of rhe others, the unique reversal and the identity are the only automorphisms. We may observe that the system with the group of order 18 is the one that was discovered by Rosa [4].
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References [ I ] J . Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1972) 315-319. [2] P.B. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. thesis, University of Toronto 1976 (=Technical Report no. 92, Department of Computer Science, University of Toronto, May 1976). [3] E.S. Kramer, Indecomposable triple systems, Discrete Math. 8 (1974) 173-180. [ I ]A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972) 61-71.
Annals of Discrete Mathematics 7 (1980) 265-276 @ North-Holland Publishing Company
A N ENUMERATION METHOD FOR NOMSOMORPHIC
COMBINATORIAL DESIGNS L.P. PETRENJUK and A.J. PETRENJUK Department of Mathematics, Institufe of Agricultural Engineering, Kirovograd, USSR
In this paper we analyse the substance of an approach to the problem of the constructive enumeration of nonisomorphic combinatorial designs. This approach consists in constructing, by means of transformations described in Section 3, of a large set of designs with a given parameter set and their subsequent pairwise comparison for isomorphism. Distinguishing between nonisomorphic designs can be done by using invariants; some of them are described in Section 2. These transformations and invariants in combination with computers form a convenient machinery when searching for new nonisomorphic designs and trying to obtain estimates for their number.
1. Basic definitions and problems Let E be a set of cardinality r, P ( E ) be the collection of all subsets of E, and P,(E) the collection of all subsets of E with cardinality r, r s n . Define on P ( E ) a function n(P) taking on values in the set Z' of integers. To this function corresponds a collection B of subsets of E in which the subset (block) occurs n(P) times. Such a collection will be called a finite incidence system (FIS) based on E, and the function n(P) will be called the block multiplicity function of this system. The value n(P) will be called the multiplicity of the block p in the system B. A finite incidence system defines o n the set P ( E ) another function
called the weight function of the system B; its value p ( X ) is called the weight of the set X in the system B. In order to indicate that n(P) and p ( X ) is the block multiplicity function and the weight function of a FIS B, they are denoted by n, ( p ) and pB ( X ) , respectively. The function n(P) determines p ( X ) uniquely; however, given a function p ( X ) , the corresponding function n(P) (and consequently, the corresponding incidence system) may not exist. Necessary and sufficient conditions for its existence (which are quite complicated) were obtained in [2]. 265
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A problem arises here whose special cases have been studied extensively. Let Q c P ( E ) , and let a function p * ( X ) be defined on Q, with values in Z'. Under what conditions does there exist a FIS B such that its block multiplicity function n ( X ) equals zero provided I X l $ K , and the weight function on Q coincides with p * ( X ) ? Here K c { l , 2 , . . . , n } . If such a FIS exists, we will call it a ( K , p*, Q, E)-design. The formulated problem will be called the existence problem for a (K, p * , Q, E)-design. The case when Q = P , ( E ) is of greatest interest. It is natural to call a (K, p*, Q, €)-design with such a Q an I-weighted design [3]. If, in addition, p * ( X )= A =constant then we have an I-balanced design [3]. 1-balanced designs were considered first by Hanani [4] who denoted them by P ( K , I, A, n ) . If ( K ( = 1, i.e. the design consists of blocks of size k then an 1-weighted ( K , p * , 0, E)-design is called a tactical configuration and is denoted by C ( k , I, A, n). Thus, a tactical configuration C ( k , I, A, n ) is a collection of blocks P I , . . . , pb (some of them may be the same subsets of the set E) such that each block contains exactly k elements and every 1-subset of E is contained in exactly A blocks. Tactical configurations with 1 = 2 are called balanced incomplete block designs. Tactical configurations with 1 = k - 1, A = 1 are called Steiner systems; if k = 3 we have Steiner triple systems (STS), if k = 4 we have Steiner quadruple systems (SQS) etc. Tactical configurations C(2, 1, A, n ) are nothing else than undirected regular graphs of degree A. Other generalizations of tactical configurations which can be studied by methods described here were defined and investigated in [l], Two FISs B and B based on sets E and E, respectively, are said to be isomorphic if there exists a 1-1 correspondence 4 : E-E such that for each block P in B, @$ = 6 implies n,@) = ne(P). Isomorphic FISs differ only by names of elements, thus the distinction between them is unessential. Isomorphism between FISs is an equivalence relation. The enumeration problem for FISs is the following: determine (or, at least, find bounds for) the total number of equivalence classes under this relation. If one is interested not in the number of equivalence classes but in a list of their representatives (one from each class), the corresponding problem is that of a constructive enumeration of nonisomorphic FISs. A natural way to solve the problem of constructive enumeration is in the following: (1) one constructs a large set of ( K , p * , 0, E)-designs, and then (2) one shows that among them there is a certain number of pairwise nonisomorphic ones. As a result, one obtains a lower bound for the number of nonisomorphic designs of this kind. This is how a lower bound for the number of nonisomorphic SQS of order 16 wa5 obtained in [19], and the number of all nonisomorphic STS of order 19 with a
An enumeration method for nonisomorphic combinatorid designs
261
head was obtained in [13]. It is not an exaggeration to say that almost all known bounds for the number of nonisomorphic FISs of various kinds have been obtained by using this kind of an approach.
2. Methods of distinguishing nonisomorphic designs A description of several methods of constructing FISs with given properties can be found in the literature (see e.g., [20]). Often a tactical configuration can be constructed in several ways. How does one determine in such a case whether the obtained designs are isomorphic or not? This problem of distinguishing and identifying is meant to be solved (and in many cases is successfully solved) by invariants. The notion of an invariant is widely used in mathematics. In general it can be defined as follows [ 5 ] . Let D and R be sets, and let c and p be equivalence relations defined on D and R, respectively. If there is a 1-1 mapping f : D + R such that dlud, implies f(d,)pf(d,) for any dl, d,E D, then f is said t o be an invariant in the set D. It follows from this definition that if for the elements d l , d, E D the relation f(dl)pf(d2) does not hold then d,cd, cannot hold either. This principle forms the basis for distinguishing through invariants. Naturally, in order to justify actually using an invariant f to distinguish elements of D with respect to an equivalence c it is necessary that certain conditions be satisfied. The simplest of them are formulated in [ 5 ] . The essence of these conditions is that the complexity of computing f(d,) and f(d2)and of testing their equivalence under p should be much smaller than the complexity of a direct testing of the equivalence of d , and d, under u. To distinguish nonisomorphic FISs it is preferable to use invariants constructed according to the following principle. One determines a class of fragments of FISs, i.e. such a collection of parts of FISs which is mapped by isomorphisms into corresponding collections of parts of the images. Elements of the basic set E, blocks, collections of blocks, collections of elements, subsystems etc. can all serve as fragments. Denote by M ( B ) a collection of fragments of a given kind of a certain FIS B. With each fragment 4 E M ( B ) we associate a characteristic c ( 4 ) which must have the property that if under an isomorphism of two FISs B and B, the image of 4 E M ( B ) is 6 E M ( B ) , then c(4)= E(6).Here c and E are characteristics of the same kind defined in M ( B ) and M ( B ) , respectively. A characteristic of a fragment reflects either properties of its internal structure or those of its position in FIS, in particular, among other fragments. A specification of a set of fragments by their characteristics represents the mentioned invariant. In this case D is the collection of all FISs, R the set of the mentioned specifications, c the isomorphism of FISs and p the equality of specifications.
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If for two FISs the values of an invariant are computed and they are different then the two FISs are nonisomorphic. If the invariants are equal then, in general, the problem of distinguishing and identifying remains unsolved, and different methods must be employed (for instance, using a different invariant). Invariant f is complete in D if from f(d,)pf(d,) always follows d,ad2. The opposite end of this ideal situation represents a triuial invariant which maps all elements of D into the same equivalence class of p in R. Let D contain exactly d equivalence classes of u,and let the invariant f take on values representing exactly r equivalence classes of p in R. The sensitivity a ( f )of the invariant f is the ratio r/d. This quantity characterizes (in a first approximation) t h e distinguishing capability of the invariant f. The sensitivity of a complete invariant is maximal and equals 1 while the sensitivity of a trivial invariant is minimal. Let us describe now an invariant used for distinguishing STSs. The main notion w e will use is that of interlacing of two elements in an STS; this notion is encountered i n Cole [6] and Cummings [7]. Let A, be an STS of order n. For each element x of the basic set E and some element y E E form the following set of pairs of elements of E:
Let now x E E, y E E (x # y). The undirected graph r,, whose vertex set is the set E and whose edge set is I7: U I7: is said t o be the graph of interlucing of elements x, v in the system A,. It follows from the properties of STS that the graph I ; , is a union of disjoint cycles o f even lengths, with length of each cycle no less than 4. The symbol Cs',!, . . . , SLY) where si and ri are natural numbers, 2 s s 1< . . .< s,,,, r,s, = $ ( r ~ 3 ) , determines the type of inferlacing of elements x, y in A, if the graph r,, contains exactly ri cycles of length 2si. Clearly, the number of distinct types of interlacing for every order of STS does not exceed the number of partitions of i ( n - 3) into parts not less than 2. For example, when n = 19, the types of interlacing are: TI = (2"). T , = (2,, 41, T3= (2, 32),Z'; = ( 2 , 6 ) , T , = (3,5), T(, = 14,4), T , = ( 8 ) . For any n. the types of interlacings can be ordered lexicographically provided one treats the symbol S ' as a string of r letters s. Number the types of interlacing in lexicographic order: T , , Tz,. . . , T,,. Then to each element x E E we may assign a p-dimensional vector
where 1, is t h e number of elements of E having type of interlacing T, ( j = 1.2, . . . , p ) with x. Obviously, t , + t 2 + - . . + tp = n - 1. We call ( f , , t2, . . . , t,,) the vector-index of X in the STSA,. This vector-index plays the r61e of t h e characteristic, and the elements of the basic set E are fragments. After the vector-indices of all elements of E in the STSA, have been
269
A n enumeration method for nonisomorphic combinatorid designs
computed, we may assign to this STS a table
T(A,,)=
. ........ ... . ... ...t(k) @)
where in each row the number lh, ih > O , is the number of elements of E having (tih)',. . . , t:)) as their vector-index in the STS A,,. For the sake of uniqueness, the rows of the table are in increasing lexicographic order. Observe that I , + * + lk =
n. The obtained table represents a specification of elements of the set E by their vector-indices. We will call it invariant table (or T-table) of the system A,,. Below are two STS of order 19 obtained by the Skolem method [8], and their T-tables. One can see that the values of the invariant are distinct, consequently these STSs are nonisomorphic. STS No. 1 0 0 0 0 0 0
1 2 3 4 6 7
0 9 0 11 0 14 1 2 1 3 1 4
5 8 10 18 17 16 12 13 15 6 9 11
1 1 1 1 1 2 2 2 2 2 2 2
7 8 10 12 15 3 4 5 9 11 13 16
18 17 13 14 16 7 10 12 18 14 15
3 3 3 3 3 3 4 4 4 4 4
17
5
4 5 6 12 14 17 5
8 11 13 15 16 18 9 6 12 7 14 13 16 15 17 6 10
5 5
5 5 6 6 6 6 7 7 7 8
7 8 14 16 7 8 9 15 8 9 10 9
13 15 17 18 11 14 16 18 12 15 17 13
8 8 9 9 10 10 11 12 13
10 11 10 11 11 12 12 13 14
16 18 14 17 15 18 16 17 18
T-table of STS No. 1 10 0 0 6 2 0
101 191
The T-table is a complete invariant in the set of STS of order 13 and 15. For orders n 3 19 as of yet there is no information on the sensitivity of the T-table as an invariant. Other invariants for distinguishing STSs are described in [9]. A description of how to use invariants to obtain information about the internal structure of STS is also contained in [9]. To distinguish between SQSs, one could use an invariant of the following type. Let an SQS (denote it by P) be based on the set E = (1, . . . ,n}. For each a E E denote by aP/aa the STS consisting of all triples (x, y, z) of elements of the set E \{a} such that (a, x, y, Z ) E P. The system aP/aa is called the derived STS of P with respect to a.
L.P. Pefrenjuk, A.J. Petrenjuk
270 STS No. 2
~~
0 0 0 0 0
1 2 3 4 5 0 6 0 13 0 14 0 15 1 2 1 3 I 4
9 7 10 8 11 12 17 16 18 10 % 11
1 1 1
1 1 2 2 2 2 2 2 1
5 7 12 15 16 3 4 5 8 12 13 17
6 13 14 17 18 11 6 9 14 16 15 18
3 3 3 3 3 3 4 4
4 5 6 12 14 15 5 7 4 12 4 13 4 I6 5 8
9 7 18 13 17 16 10 18 15 14 17 18
5 5 5 6
6 6 6 6 7 7 7 7
12 13 14 7 8
17 16 15 15 13 9 16 10 14 11 17 8 16 9 14 10 17 1 1 12
8 8 8 9 9 9 10 10 11
9 10 11 10 I1 12 11 13 14
17 12 15 15
13 18 16 18 18
T-table of STS No. 2 I 8 I 1 2 2 11 3 10 4 6 5 6 6 0
6 1 0 2 3 2 I)
1 1 2 1 2 2 2 1 3 2 2 3 0 3
1 1 1 0 0 0 0
0 0 0 0 0 0 0
3 3 3 3 3 3 I
-
Consider the collection of tables
If the table 7‘’’ oxcurs among them a , times, T”’ a2 times etc. then the system P is assigned a symbol-specification
d ( P ) = u , T ” + a , T Z ’ +. . where the ”symbolic” summation is commutative. It is easy to see that d ( P ) is an invariant in the set of SQSs. W e will call it the derivative invariant (or derivant) of the system P. O n request o f one of the authors, S.G. Buzdugan has written a program to obtain derivants of SQSs of order 16. H e found 41 SQSs of order 16 having distinct invariants. This suggests good distinguishing possibilities of the derivant*. The following generalization of this invariant seems interesting. Let A be a tactical configuration C(k. I, A, n j based on E and let X c E, = t where 1 d t < 1. Denote by dA/dX the FIS based on E \ X and determined by the multiplicity function o f blocks
1x1
r t ( P ) = n*iP
u X).
* Editor’s romrnenf. The invariant d ( P ) does not distinguish the two nanisornorphic transitive SOS of order 16 based on PG(3.2) (see. e.g. P.B. Gibbons, R.A. Mathon, D.G. Corneil, Steiner quadruple \wtern\ o n 16 symbols. Proc. 6th S.-E. Conference Comhinatoria. Crraph Theory and Computing, H O C ~Raton. 1075. pp. 345-365).
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271
The system dA/dX is a tactical configuration C(k - r, 1 - t, A, n - r) and is called derivative of A with respect to X. The system aP/aa considered above is a special case of such a derivative for 1 x1= 1. Let further I be an invariant on the set of tactical configurations C(k- r, 1 f, A, n - t ) which takes on values from a finite set { I , , . .. , I N } . Consider the collection of configurations {dA/dX,X c E, 1 x1 = 1). If for a1 among them the invariant I takes on the value I , for a2 the value I , etc. then the "symbolic" sum d,(A)=alIl+(~212+* * * +aNZN
is an invariant in the set of tactical configurations C(k, 1, A, n). Obviously a1+ . . .+a, =(Y). The invariant d,(A) will be called the (f, I)-derivant of the system A. The derivant d(P) is thus a (1,t)-derivant of the SQS P. The following invariant penetrates even more deeply into the structure of a tactical configuration. For each S, S c E, IS(= s, O < s < t, let there exist & ( S ) sets X, X c E, X =) S, 1 x1= r such that dA/dX = Zj ( j = 1,2,. ... N ) . Denote 4 ( S )= I?==, pj(S)Zi.If the subsets of S are considered fragments and 4 ( S ) their characteristics then the specification of the set P,(E) by 4 ( S ) is an invariant in the set of tactical configurations C(k, I, A, n ) . Call it (t, I),-derivant of the configuration A, and denote it by d:(A). The (f, I),-derivant can be represented by a table
K I p:"
dh(A)=
p:". . .
.................. p y ...
pi"'
where
E,
is the number of sets S, S c E, IS1 = s such that
4 ( s ) = p : " I , + p ~ ) ~ 2 " " + p ~ ) I N , Y = 1 , 2 , . .. ,w. The rows of the table are in increasing lexicographic order. Let us describe how to construct another invariant. Let B be a FIS based on E, and let b be the number of its blocks. For each of the (!) unordered collections of I blocks form the intersection vector (xl,x2.. . . . x), where is the number of pairs of blocks p, p' in the considered collection such that Ip n p'( = i ( i = 1,2, . . . . n). Form a table
where xy) + x y ' + . . . + x(+')= (,I, ' h , + . . + h, = (p), and h, is the number of collections of t blocks having as their intersection vector (x'i"', . . . ,x!,")), The rows of the table are in increasintg lexicographic order.
L.P. Petrenjuk, A.J. Petrenjuk
272
This table is also an isomorphism invariant in the set of FISs. Call it the rr -invariant.
Clearly, due to the complexity of computations, computing invariants like the ones described above without an aid of a computer is not feasible. Invariants based on the same principle are described and have been used successfully to distinguish nonisomorphic objects in [ 5 , 9 , 11-18].
3. Construction of weighted HSs by using mutually balanced collections of blocks Two FISs A and B both based on the set E are mutually balanced on
Q, Q = P(E) if PA
(x)= p B (x)
for all X E 0. Using mutually balanced FISs one can construct from a given (K, p * , Q, E)-design other (K, p*, Q, E)-designs that are, in general, not isomorphic t o the design one starts with. In order to show this let us introduce some relations and operations on FISs. We will write A s B provided n A ( P ) S n B ( P for ) all PEP(E). The sum A + B of two FISs based on E is a FIS with the block multiplicity function n A t L ) ( 6 )= nA (0) + nB (0). The difference A - B is defined to be the FIS whose block multiplicity function is
The intersection of two FISs A and B is defined to be the FIS A n B with the block multiplicity function nAnB
(PI = min (nA( P ) , nB(P))-
Theorem 1. Let H be a ( K , p*. 0, E)-design, A and B be FISs based on E and mutually balanced on Q, and A c H . Then (H - A ) + B is a ( K , p*, Q, El-design.
the summation here is over the set
(0 : 0 2 X).
Theorem 2. Let H and G be two (K,p*, Q, E)-designs. There exist FISs A and B based on E and mutually balanced on Q such that ( H - A ) + € ?= G.
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273
Proof. It is easy to see that A = H - G and B = G - H are just those FISs whose existence is asserted in the statement of the theorem. The transition from the FIS H to ( H - A ) + B performed in Theorem 1 is called a substitution in H of the collection of blocks A by a collection of blocks B that is mutually balanced with A on Q. The essence of the two theorems is in that we can obtain from a ( K , p*, Q, E)-design H any other (K, p*, Q, E)-design by an appropriate substitution. Observe that to achieve this end it suffices to consider only pairs of mutually balanced collections of blocks A, B such that A r l B = 9. A special case of the substitution of blocks is the d-transformation of graphs described in [101. For STS the notion of substitution of collections was introduced in [9], and in general form this notion appeared in [18]. The question arises, how to find pairs A, B of mutually balanced collections of blocks on Q such that one of these collections is contained in H? We will suggest a construction method for such pairs of collections for the case of tactical configurations C(k, I, 1, n ) . Let H be a configuration based on E and let x E E, y E E, x f y . Denote by aH/axI, the FIS obtained as a result of deleting from aH/dx of all blocks containing y . Clearly, dH/dx), and aH/ay, 1 are mutually balanced collections of blocks on P[-,(E). Consider a bipartite graph r,, whose vertices are the blocks of these collections, and edges are joining two blocks from different collections if they have an ( 1 - 1)-subset in common. Suppose this graph has m ( m5 1) components. Consider a subgraph r of the graph r,, which is a union of several components. Denote by M and N, respectively, the collection of all those blocks from aH/ax(, and aH/ayl,, respectively, that are vertices of the graph Clearly, the collection of blocks
r.
is contained in H , and the collection
N = { p : P = f l ’ U { y } , p ’ ~ M } U { p: P = P ’ U { X } , ~ ’ E N } is mutually balanced with on Pl(E). ( H - fi)+ N is a tactical configuration different from H. Let us call a substitution of collection of blocks performed according to the way described above an X-substitution. Observe that to determine in H a collection M and to construct the collection 13 that is mutually balanced with it depends on the choice of elements x , y and on the choice of the components that determine the graph r. In the degenerate case when r,, [email protected] described collections are undefined. Consider an example. Let H denote the STS No. 1 considered earlier. Write in columns the sets of pairs aH/a3 and dH/a5.
L.P. Petrenjuk, A.J. Petrenjuk
274
a~ i a 3
awas
5
11
3
11
12 1s 2 7 6 13
0 4 8 2 7 6
1 9 15 12 13 10
17 18
Below the horizontal line are the sets of pairs aH/a3Is and aH/d5)3.Between the columns is the diagram of the graph r3.5. If we take now for r the larger component of the graph r3.5 then the described method yields the following mutually balanced collections of triples: -
M 0 1 3 3 2 3
N
3 10 3 9 4 8 12 15 3 7 6 13
0
1
4 5 2 5
5 9 8 1s 5 12 7 13 6 10
5
5
0 5 10 1 5 9 4 5 8 5 12 15 2 5 7 5 6 13 0 1 3 3 4 9 3 8 15 2 3 12 3 7 13 3 6 10
Thus, we defined a collection of transformations each of which transforms the tactical configuration C ( k ,I, 1, n ) into a tactical configuration based on the same set and having the same parameter set but having its block structure different from that of the original configuration. If we take into account that on tactical configurations obtained from H we can perform X-substitutions of collections of blocks as well then it will become clear that this construction method can produce a large family of tactical configurations C(k,I, 1, n). Is it possible for any two classes of isomorphic configurations C(k, I, 1, n) to form a finite sequence of X-substitutions of mutually balanced collections of
An enumeration method for nonisomorphic combinatonal designs
275
blocks that transforms representative of one of these classes in some representative of the other class? This is the completeness problem of the constructed family of transformations. It can be formulated as follows. Two tactical configurations C(k, I, 1, n ) are reachable if there exists a sequence of X-substitutions which effects a transformation of one of them into a configuration isomorphic to the other one. Reachability is an equivalence on the set of configurations C(k, I, 1, n ) and the question is, what is the number r = r ( k , I, 1, n) of its equivalence classes. If there is only one such class then knowing a single tactical configuration C(k, I, 1, n ) one can obtain through X-substitutions a family containing representatives of all isomorphism classes of the set of tactical configurations having these given parameters. If r > 1 then for such a construction we need a basis consisting of r pairwise unreachable tactical configurations C ( k ,I, 1, n). One has r ( 3 , 2 , 1, n ) = 1 for n = 3 , 7 , 9 , 13 (the corresponding X-substitution in the case n = 13 can be found in [6]) and this leads one to the following conjecture: r(k, I, 1, n ) = 1 for all parameter sets k , I, n such that the corresponding configuration exists. On the other hand, it should be observed that in a similar problem on H-transformations of 1-factorizations of order n (see [ll]) already for n = 8 and 10 the number of corresponding reachability classes equals 2, and for n = 12 it is not less than 6 [12].
References [l] P, Hell and A. Rosa, Graph decompositions, handcuffed prisoners and balanced P-designs, Discrete Math. 2 (1972) 2 2 9 2 5 2 . [2] A.J. Petrenjuk, Kriterii realizujemosti funkcii kak vesovoi funkcii koneEnoi sistemy incidencii, Vestnik Mosk. Univ. Ser. Mat. Meh. 1971, No. 4, pp. 16-20. [3] A.J. Petrenjuk, 0 suEestvovanii vzveiennych koneEnyh incidentnyh struktur, Doklady AN SSSR 193 (1970) 535-536. [4] H. Hanani, On some tactical configurations, Canad. J. Math. 15 (1963) 702-722. [5] A.J. Petrenjuk, Primenenie invariantov v kombinatornych issledovaniach, in: Voprosy Kibernetiki, Trudy Seminara po Kombinatornoi matematike, Moskva, Sov. Radio 1973, pp. 129-136. [6] F.N. Cole, The triad systems of thirteen letters, Trans. Amer. Math. Soc. 14 (1913) 1-5. [7] L.D. Cummings, On a method of comparison for triple systems, Trans. Amer. Math. Soc. 15 (1914) 311-327. [8] T. Skolem, Some remarks on the triple systems of Steiner, Math. Scand. 6 (1958) 273-280. [9] A.J. Petrenjuk, Priznaki neizomorfnosti sistem trojek Stejnera, Ukr. Mat. 2. 24 (1972) 772-780; English translation: Tests for nonisomorphic Steiner triple systems, Ukr. Mat. Zh. 24 (1972) 620-626. [lo] S. Hakimi, O n realizability of a set of integers as degrees of the vertices of graph, J. SIAM 10 (1962) 496-506. [ll] V.V. Voznjak and A.J. Petrenjuk, Ob odnom algoritme peretislenija sistem grupp par, in: Kombinatornyj Analiz, Vyp. 2 pp. 38-41, [ 121 L.P. Petrenjuk and A.J. Petrenjuk, 0 perecislenii soverknnyh 1-faktorizacii polnyh grafov (to appear). [ 131 I.P. Neporoinev and A.J. Petrenjuk, Konstruktivnoje peretislenije sistem grupp par i oglavlennyje sistemy trojek Stejners, I. Kombinatornyj Analiz 2 (1972) 17-37; 11. 3 (1974) 28-42; 111, t o appear.
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[ 141 L.P. Petrenjuk and A.J. Petrenjuk, 0 konstruktivnom perefislenii 12-ver5innyh kubikskih yafov, Kornbinatornyj Analiz 3 (1974) 72-82. [ 151 L.P Petrenjuk and A.J. Petrenjuk, 0 perefislenii desiativerginnyh odnorodnyh grafov stepeni 4, in: Voprosy Kibernetiki. Trudy Serninara po Kombinatornoi matematike, Moskva, Sov. Radio 1975 1161 L.P. Petrenjuk and A.J. Petrenjuk, Postrojenie nekotoryh klassov kubiteskih grafov i neizornorfnost' Kirkmanovyh sistem trojek, Kombinatornyj Analiz 4 ( 1976) 73-77. [17] Kh.J. Kurek and A.J. Petrenjuk, 0 pokrytii grafov zviozdami, in: Teorija Grafov. Kiev (1977) 145-156. [ 181 A.J. Petrenjuk. lssledovanija v teorii konefnyh sistem incidentnostei, Cand. Diss., Moskva, 1971. [191 C.C. Lindner and A. Rosa, There are at least 31,021 nonisomorphic Steiner quadruple systems of order 16. Utilitas Math. 10 (1976) 61-64. [20] J Doyen and A. Rosa. A bibliography and survey of Steiner systems. Bollet. Unione Mat. Ital. (4) 7 (1973) 392-419.
Annals of Discrete Mathematics 7 (1980) 277-300 @ North-Holland Publishing Company
ON CYCLIC STEINER SYSTEMS S(3,4,20) K.T. PHELPS School of Mathematics, Georgia Institute of Technology Atlanta. G A 30332, USA A Steiner quadruple system of order n is said to be cyclic if it has an n-cycle as an automorphism. In this paper we enumerate all cyclic Steiner quadruple systems of order 20. As a necessary prelude t o this enumeration, a number of results are established, including a new characterization of the existence problem in terms of hypergraphs. It is also shown that the necessary conditions for an S-cyclic SQS (n) is that n = 2,4,10 or 20 mod 24, and that there does exist an infinite class of cyclic Steiner quadruple systems, thus answering a question posed by Lindner and Rosa [16].
1. Introduction A Steiner system S(3,4, n), more often called a Steiner quadruple system of
order n (briefly SQS (n)) exists for all n = 2 or 4 mod 16. In general a Steiner system of order n is said to be cyclic if it has an automorphism consisting of a single n-cycle. A cyclic Steiner System S(t, k, n), consisting of an n-set P and a collection B, of k-element subsets of P, can thus be assumed to have as its n-set Z,,, the residues m o d n and (Z,,,+>, the integers mod n under addition as a subgroup of its automorphism group. The existence of cyclic Steiner triple systems was settled by R. Peltesohn [17]. For Steiner quadruple systems, this question is unresolved. In a previous computer investigation, Guregovii and Rosa [12] have enumerated all cyclic SQS (n)for n s 16: they established that no cyclic SQS (n)exists for n = 8, 14 or 16 and exactly one exists for n = 2, 4 and 10. Others have established the existence of cyclic SQS (n) for various values of n (e.g. [8, 161). Until recently (Phelps [19]), the only known cyclic S Q S ( n ) were S-cyclic (for a definition of S-cyclic see Section 3 below). Jain [14] constructed the first S-cyclic SQS(20) and showed that it was the unique S-cyclic system. Later Phelps [19] constructed another cyclic SQS (20). More recently Griggs and Grannell [ll] and Cho [3] have constructed other examples of cyclic SQS (20). With these diverse results in mind it appeared that a complete enumeration of cyclic SQS (20) would be especially warranted. A primary purpose of this paper then is to establish that there are exactly 29 nonisomorphic cyclic SQS (20), thus extending the results of Guregovii and Rosa r121. A complete enumeration of cylic SQS(20) given limited resources, did not appear feasible at first. To solve this enumeration problem, a better understanding 277
K.T. Phelps
218
of the general existence problem had t o be developed. This mathematical basis along with a proper characterization of this problem, enabled the author to enumerate all cyclic SQS(20) with only a modest utilization of computer resources (less than an hour of computer time on a CDC Cyber 70). Thus it is felt that the approach used, that is the algorithmic methods and the general results upon which they are based is almost as interesting as the results that were obtained. In the next section, the general existence problem for cyclic SQS(N) is discussed. Some new results are presented. Having established the mathematical background, we then proceed t o discuss the methods used and finally the results that were obtained. For more on cyclic quadruple systems the reader is referred to an excellent survey article by Lindner and Rosa [16].
2. Background Let Pr(Z,,)be the collection of all r-element subsets of Z,, = {O, 1 , 2, . . . , n - 1). To each triple {i. j , k} in P3(Zn). i < j < k, we can associate a difference triple ( j - i, k - j , i - k) with the differences taken mod n. Two difference triples are equivalent if and only if one is a cyclic re-ordering of the other. Under the action o f (Z,,,+), two triples of CPJZ,) will be in the same orbit if and only if the associated difference triples are equivalent. For example, all distinct difference triples mod 20 are listed in Table B.l.Note that for n = 2 or 4 mod 6 every orbit in 9,(Z,) is full (i.e. of size n). Table 6 . I . Difference triple mod 20 I
1
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i n a similar manner, one can associate a difference quadruple (a, b, c, d ) with each orbit in S,(Z,,). Again a quadruple {i, j , k , I } will be associated with the difference quadruple ( j - i, k - j , 1 - k. i - I ) where the differences are taken mod n. One also has that the four 3-element subsets of a quadruple will give rise to four (not necessarily distinct) difference triples and these in turn characterize the orbit of the quadruple; that is, two quadruples will be in the same orbit if and only if they have the same difference triples. Let the difference quadruple (a, b, c, d ) be Then (a, b, n - a - b ) , (b, c, n - b - c ) , associated with a particular orbit in P4(Z,,).
On cyclic Steiner systems S(3,4,20)
279
(c, d , n - c - d ) and ( d , a , n - a - d ) will be the four difference triples that characterize the orbit. Since our interest is in cyclic Steiner quadruple systems we are only concerned with those quadruples that can occur in some such system. It is not too hard to see that for such a quadruple, the number of non-equivalent difference triples it contains must be either one, two or four. Under the action of (Z,,, +), a quadruple will have a short orbit if and only if the number of distinct difference triples is one or two. For example, the quadruple {1,6, 11, 16) having difference quadruple ( 5 , 5 , 5 , 5 ) mod 20, has only one distinct difference triple ( 5 , 5 , 10) and thus has a short orbit of size 5. Remembering that in a difference quadruple ( a , b, c, d ) mod n, a + b + c + d = n and n = 2 or 4 mod 6, it is a simple exercise to show:
Lemma 2.1. The difference quadruple ( k , k, k, k ) , where n = 4k, is the only difference quadruple having exactly one difference triple and hence there can be ( a t most) one orbit of size i n . Lemma 2.2. The difference quadruples ( i , j , i, j ) (mod n ) , i f j , i + j = i n are the only difference quadruples having exactly two distinct difference triples and hence the corresponding orbits have size 2n. Above, it was claimed that orbits are characterized by their difference triples. Since it will be important later we restate and prove thus
Lemma 2.3. T w o differencequadruples giving rise to the same difference triples are equivalent.
Proof. If the difference quadruples have only one or two distinct difference triples then using Lemmas 2.1 and 2.2 the statement is clearly true. Assume that there are four distinct difference triples. If two triples are subsets of the same quadruple, they must intersect in a pair and hence the associated difference triples must have at least one difference in common. If the difference triples have exactly one difference in common then there can only be one way these difference triples can be contained in the same quadruple. It is easy to see that every quadruple must have at least 2 such difference triples among its 4 difference triples. The lemma follows immediately from this observation. Note that a pair of difference triples can be contained in more than one orbit or rather more than one difference quadruple only if the difference triples have more than one difference in common (e.g. ( i , j , k ) and ( j , i, k)). With Lemma 2.3, we have 3 ways of characterizing the orbits of a cyclic SQS (n): we can choose a quadruple from each orbit -such quadruples are usually called base blocks; also associated with each orbit is a difference quadruple; finally with each orbit we can associate a set of difference triples. If we
2x0
K.T. Phelps
number the difference triples and consider them as vertices of a hypergraph with the edges corresponding to different orbits, then a cyclic SQS (n) will correspond to a 1-factor or perfect matching in this hypergraph (see Berge [l] for undefined terms). For example Table B.3 lists all difference quadruples (mod20) which correspond to full orbits. Table B.4 lists the 4-sets of difference triples for these orbits.
3. Isomorphisms and automorphisms of cycle SQS (n) When a permutation group acts regularly o n a set the automorphisms of that group induce (additional) permutations on this set. In the case of (Z,,+),its automorphisms are merely the units of the ring of integers mod n and the induced permutations are of the form p : x + ax mod n, where a and n are relatively prime. If we have a cyclic S Q S ( n ) then a unit of 2, will either induce an additional automorphism of the quadruple system or it will give a distinct (isomorphic) cyclic SQS (n). Let us consider the action of the units of the ring of integers mod n more carefully. The units act as automorphisms of 9,(Zn).In particular, these automorphisms will permute the orbits of PJZ,,) under (Z,,, +) and hence will induce a permutation of the difference triples. Consider the permutation p :x + - x mod n induced by the unit n - 1. A triple {O, i, i + x} where i + x < n gives us the difference triple ( i , x, n - i - x ) mod n. The permutation p maps (0, i, i + x} into (0, n - i, n - - i - x } or rather (0, n - i - x, n - i } which gives us the difference triple (x, i, n - i - x). Each unit then gives us a permutation of Z, and induces a permutation of the difference triples. All units mod 20 and their induced permutations are listed in Table B.2. In a similar fashion these units induce permutations of the difference quadruples. The above observations can be very useful. For instance, a difference quadruple ( a , b, c, d ) is said to be symmetric if for some cyclic re-ordering we have either (1) a = b and c = d, or (2) a = c (see [ 161). A cyclic SQS ( n ) is said to be S-cyclic if all of its difference quadruples are symmetric. It is simple to show that the permutation 0 : x + - x discussed above fixes symmetric difference quadruples. Hence we have:
Lemma 3.1. An S-cyclic S Q S ( n ) always has p : x + - x automorphisrn.
mod n as an addition
Since units always fix 0 and i n , these permutations can be considered as automorphisms of the derived triple system as well. Corollary 3.2. A necessary condition for an S-cyclic S Q S ( n ) to exist is that
n = 2 , 4 , 1 0 or 20 mod 24.
281
On cyclic Steiner systems S(3,4,20) Table 8.2. Units mod 20 (1)
-
unit 1 3
permutation:
(1,13,9,17) (4,12,16,8)
(2,6,18,14)
(3,19,7,11)
induced permutation of difference triples: (1,54,19,23) (6,17,26,11) (21,55,37,29) (2)
unit 3
-
(2,50,56,52) (3,7,39,28) (4,48,44,25)(5,42,46,22) (9,27,10,30) (13,41,15,49) (14,16,32,51) (18,40,20,36) (31,47,35,43) (8,33) (12,531 (24,571 (34,38) ( 4 5 )
permutation:
(1,3,9,7) (2,6,18,14) (4,12,16,8) (5,15) (11,13,19,17)
induced permutation of difference triples: (lr23,19,54) (7r21,28,37) (22,49,42,41) (3)
unit 9
-
(2,26,56,6) (3,29,39,55) (4,43,44,47) (5,13,46,15) (9,27,10,30) (11,52,17,50) (14,16,32,51)(18,40,20,36) (25,35,48,31) (8,241 (12,53) (33,57) ( 3 4 , 3 8 ) ( 4 5 )
permuation:
(1,9) (2,18) (3,7) (4,16) (6,14) (8,12) (11,19) (13,17)
induced permutation of difference triplee: (1,191 (2,561 (3,391 (4,441 (5,461 (6,261 (7,28)(9,10)(11,17)(13,15) (14,321 (16,51) (18,20) (21,371 (22,42) (23,541 (25,48) (27,30) ( 2 9 , 5 5 ) (31,351 (36,401 (41,49) (43,471 (50,521 ( 8 ) ( 1 2 ) ( 2 5 ) ( 3 3 ) ( 3 4 ) ( 3 8 ) ( 1 5 ) (53) (57) (4)
unit 1 9
-
permutation:
(1,191 (2,18) (3,171 (4,161 ( 5 , 1 5 ) (6,14) (7,131 (8,121 (9,11)
induced permutation of difference triples: ( 2 . 1 1 ) ( 3 , 2 1 ) (4,311 ( 5 , 4 1 ) ( 6 , 5 0 ) (7,551 (8,571 ( 9 . 1 0 ) (13,221 ( 1 4 , 3 2 ) ( 1 5 , 4 2 ) ( 1 6 3 1 ) ( 1 7 , 5 6 ) (18,201 ( 2 4 , 3 3 1 (25,431 (26,521 (27,301 (28,291 (35,441 ( 3 6 . 4 0 ) ( 3 7 , 3 9 ) (46,491 (1)( 1 2 ) ( 1 9 ) ( 2 3 ) ( 3 4 ) ( 3 8 ) ( 4 5 ) ( 5 3 ) ( 5 4 ) (5)
unit 11-permutation: (1,111 (3,131 (5,151 (7,171 (9,191 induced permutation of difference tirples: (1,191 ( 2 , 1 7 1 ( 3 , 3 7 ) (4,351 (5,491 (6,521 ( 7 , 2 9 ) (8,57) (11,561 ( 1 3 , 4 2 ) ( 1 5 , 2 2 1 (21,391 (23,54) (24,331 (25,471 ( 2 6 , 5 0 1 ( 2 8 , 5 5 ) (31,441 ( 4 1 , 4 6 1 ( 4 3 , 4 8 ) ( 9 ) (10) ( 1 2 ) ( 1 4 ) ( 1 6 ) ( 1 8 ) ( 2 0 ) ( 2 7 ) ( 3 0 ) ( 3 2 ) ( 3 4 ) ( 3 6 ) ( 3 8 ) ( 4 0 ) (45) (51) (53)
Proof. An S-cyclic SQS (n) has p : x + - x as an additional automorphism. This means the derived triple system of an S-cyclic SQS (n) must be a reverse triple system and these only exist for n - 1 = 1 , 3 , 9 or 19 mod 24 (see Doyen [ 6 ] ,Rosa [20], Teirlinck [21]). The above observations lead to two interesting problems: determining the spectrum for Steiner triple systems having various fixed point automorphisms (similar to those induced by the units mod n) and determining sufficient conditions for a reverse triple system to be the derived triple system of a cyclic quadruple system. The next question is, are the units the only additional automorphisms of or isomorphisms between cyclic SQS (n)? With respect to automorphisms the answer is no. The cyclic SQS (10) has the group of units mod 10 as additional automorphisms but it also has the projective general linear group PGL (2,9) as its automorphism group. This is not an isolated case.
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Table B.4. Edges of Hypergraph 1 2 312 1 7 817 2 4 12 22 2 9 17 27 3 5 22 32 3 10 27 37 4 6 32 42 4 38 48 55 5 9 40 44 6 7 50 55 6 53 54 55 7 9 20 56 7 41 48 54 8 10 19 20 8 31 39 48 9 36 37 5 0 2 1 1 13 22 1.0 1.1 30 56 I2 16 18 36 13 14 21 41 13 20 27 47 14 1.7 33 52 14 38 51. 5:3 1.5 18 30 43 15 3:’ 37 49 16 4 7 49 56 17 20 28 30 17 22 ?Y 47 1 ?1 2? 31 10 ?1 40 52 19 2 2 49 92 ? 3 :?6 ? 9 53
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31 ( 38 ( 21 ( 8 37 ( 8 11 ( 9 21 ( 83) 4 11 ( 88) 12 13 ( 93) 12 18 ( 9 8 ) 13 16 (1.03) 13 29 (108) 14 19 (113) 14 32 ( 1 1 8 ) 15 46 (123) 16 18 (1:!8) 16 32 (133) 17 46 (138) 18 36 (143) 3 21 (148) 1 2 22 (153) 22 23 (158) 23 28 (163) 24 27 (168) 14 38 (175) 25 29 (178) 19 26 (183) 26 33 (188) 3 31 (193; 19 3 2 (198) 2Y 35 ( 2 0 3 ) ‘0 35 ( 2 0 8 ) I 41 (213) 21 45 (
514 10 19 14 24 29 57 24 34 39 55 34 44 41 50 54 55 30 52 37 46 39 57 28 37 38 55 19 28 30 39 15 24 15 33 20 38 23 43 49 51 35 49 36 40 54 56 :?O 5 l 38 53 48 51 38 51 24 33 24 42 25 51 52 53 30 33 43 47 38 44 28 51 .39 54 34 43 44 46 34 37 36 41 42 50 44 57
( (
4) 9)
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14) 19) 24) ( 29) ( 34) ( 39) ( 44) ( 49) ( 54) ( 59) ( 64) ( 69) ( 74) 79) 9 ( 84) 5 ( 89) 12 ( 94) 12 ( 99) 13 (104) 14 (109) 14 (114) 15 (119) 15 (124) 16 (129) 16 (134) 17 (139) 18 (144) 4 (149) 13 (154) 23 (159) 23 (164) 24 (169) 24 (174) 25 (179) 26 (184) 27 (189) 4 (1Y4) 20 (199) 31 (204) 19 (209) 2 (214) 1 ( ( (
5 615 10 20 57 7 15 25 20 30 55 8 25 35 30 40 50 10 36 46 7 42 51 48 50 53 10 39 40 21 27 36 47 48 55 1 1 18 27 46 47 50 18 19 57 1.1 20 29 1 1 16 25 14 16 34 19 39 56 17 24 44 15 31 50 20 36 53 16 41 55 47 51 54 19 29 52 22 28 46 42 49 54 32 40 53 21 25 34 22 25 43 24 27 40 29 43 46 28 34 39 33 37 39 46 52 53 28 29 52 36 39 52 31 35 44 32 34 36 34 35 57 35 37 42 41 43 51 50 51 55
5)
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1 2
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4 5 5
6 7 7 8
8 9 ( 80) 1 6 ( 85) ( 90) 12 ( 95) 12 (100) 13 (105) 14 (110) 14 (115) 15 (120) 15 (125) 16 (130) 10 (135) 17 (140) 18 (145) 5 150) 14 155) 23 160) 33 165) 24 170) 15 175) 25 180) 26 185) 27 190) 1 1 195) 21 200) s4 ?05) 28 210) 3 215) 2
6 716 3 1 1 21 8 16 26 4 21 31 9 26 36 5 31 41 37 47 57 8 43 52 41 46 49 48 49 57 8 55 57 50 53 54 9 10 57 41 49 53 27 2a 55 1 1 12 21 1 1 17 26 15 17 35 20 40 51 19 26 46 16 32 51 37 54 56 17 42 56 42 47 48 20 38 40 17 18 55 32 39 54 22 30 48 21 26 35 72 26 44 25 38 49 30 33 36 2Y 35 48 26 41 57 4.4 47 48 30 37 40 33 40 54 32 33 50 33 34 55 35 39 47 35 38 43 41 44 5:’ 50 52 56
284
K.T.Phelps
Lemma 3.3. There exists a cyclic SQS (3' other than the units mod n.
+ l ) , for all k > 0 having automorphisms
Proof. PGL (2, q ) always has a cyclic subgroup of order q + 1 (Huppert [13, p. 1873). There exists a S(3,4,3' + 1) having PGL (2, 3 k ) as its automorphism group C2, 161. As for isomorphisms, the question remains open. Among the cyclic SQS (20) the only isomorphisms are those induced by the group of units mod20. Both these questions will be discussed further in conjunction with the presentation of the enumeration results.
4. The enumeration of cyclic SQS (20) As was previously mentioned, enumerating cyclic SQS ( n ) is equivalent to finding all 1-factors or perfect matchings in the associated hypergraph. More specifically consider the (valid) orbits of quadruples of 2,". To each orbit we can assign a subset of X, the set of all differences triples mod 20. Letting 8 denote the collection of these subsets of X we have that H(Z,,) = ( X , 8)is a hypergraph. If we find a l-factor in ( X , %), then the union of the corresponding orbits will be a cyclic SQS (20) since every difference triple is in exactly one edge and hence every triple must occur in exactly one quadruple. The units of Z,,,, induce automorphisms of the hypergraph H(Z,,). These automorphisms acting on the edges of H ( Z 2 J can be used to simplify the enumeration search. For example having found all l-factors that contain a particular edge, then one can eliminate that edge and all edges in its orbit from further consideration. Also having found a particular partial l-factor, the subgroup of these automorphisms that fixes this partial 1-factor can be used in a similar fashion to simplify the remaining search. Every cyclic SQS (20) must contain a short orbit of size 5 and hence, by Lemma 7,. 1, must contain the orbit having ( 5 , 5 , 5 , S) as its difference quadruple. That is, it must contain the base hlock {1,6, 11, 16). Thus the difference triple ( 5 , S , 10) is accounted for and hence this vertex (45) and all edges incident to it can be eliminated from the hypergraph H(Z,,J. In what follows, it will always be assumed, that we are working with this reduced hypergraph. Besides the short orbit of size 5, a cyclic SQS (20) can contain 0.2, or 4 short orbits of size 10. From Lemma 2.2 we know that these short orbits have these diKerence quadruples:
( 1 , 9, 1,9), (3.4,3.7).
(2, tA2.8).
(4,6,4.6).
In attacking this problem then we divide it into 3 cases, I. 11 and 111 in which we enumerate all cyclic SQS (20) having 0 , 2 and 4 orbits of size 10 respectively. In Case 11, there are (j) possible choices of 2 such short orbits. However this
On cyclic Steiner systems S(3.4,20)
285
collection of six choices is reduced to 4 possible subcases (namely A, B, C, D) by using the automorphlsms induced by the units of Z20. Furthermore every 1-factor must contain an edge which covers vertex 12 (i.e. difference triple #12). The degree of 12 in this hypergraph is at most 15 (depending upon the case). However, these edges fall into only 6 orbits, having edges numbered ( l ) , (‘ll), (88), (89), (91) and (93) as representatives (see Table B.4). Thus the other 9 edges can be eliminated without any loss. Moreover having enumerated all 1-factors containing one of these edges we can eliminate the remaining edges in its orbit from further consideration. The enumeration, then, consisted of a series of computer runs in which the program found all 1-factors in a suitably reduced hypergraph. Finding all perfect matchings in a k-uniform hypergraph (k >2) is in general “hard” problem (Garey and Johnson [9]).However, properties of the hypergraph are such that there is a simple algorithm that is rather efficient. First we describe the algorithm and then discuss why it is efficient. The algorithm is a simple backtrack search using many known techniques (see Gibbons [lo]). Assuming that (X, 8) is the uniform hypergraph in question, we have that at level i of the search F ( i ) contains the partial 1-factor; T(i), T ( i )G X , contains the vertices covered by the edges in F(i)and S ( i ) , S ( i ) s 8,contains the edges of the sub-hypergraph induced by the vertices of X\T(i). Let u ( i ) be a vertex of minimum degree in (X\T(i), S ( i ) ) . If the degree of u(i) is zero then this partial 1-factor can not be completed so we backtrack. If the degree of u ( i ) is nonzero we choose an edge, E E S ( i ) such that u ( i )E E. Then F(i + 1)= F ( i )U { E } , T ( i + l ) = T ( i ) U E and S ( i + l ) = { E r I E r ~ S (and i ) E’nE=@}.Since we know exactly how many edges are needed we backtrack if IS(i + 1)1C k - IF(i + 1)1, where k is the number of edges in a 1-factor. An outline of the algorithm follows: Initialize: F(O), S(O), T(O),k, i = 0. Do while i 3 0 Next-Level: Compute u(i). If degree of u ( i ) < O then go to Back-Track. Next-Edge: Find E c S ( i ) such that u(i)EE. If no such edge exists then go to Back-Track. Set F ( i + l ) = F ( i ) U { E } ; T ( i + l ) = T(i)UE. Compute S(i + 1). i=i+l. If ( S ( i ) (< k - IF(i)l then go to Back-Track. If i < k then go to Next-Level. Print F( i). Back-Track i = i - 1. S(i) = S(i)\F(i + 1). Go to Next-Edge. End; End;
K.T. Phelps
286
Using this simple algorithm the longest run on any one case was about a minute. Case I took about 4 minutes in total. T h e majority of the runs were completed in 10-25 seconds. Of course, this is partly due to the pre-run analysis, discussed above, which simplified the search. Others reasons for this efficiency lie in t h e nature of the hypergraph. Note, that in this algorithm the number of steps is dependent on the minimum degree at each level. That is if d ( i ) is the maximum degree of any vertex u ( i ) during the entire search then the number of steps is less then d ( i ) where k depends upon the case being considered. Looking at H(Z,,,) (see Table B.4) any vertex has degree between 14 and 17. Moreover any 2 vertices are contained in at most 2 edges. In fact the majority of pairs of vertices are contained in at most one edge. T h u s the minimum degree drops from one level to the next. In fact for the first couple of levels. t h e degree of almost every vertex drops and the minimum degree usually drops by at least 3. Hence the upper bound o n the number of steps, d(i). is relatively modest (e.g. 14.11 . S * 6 * ( 4 ! as ) compared to 14!). Furthermore backtracking (because the minimum degree at that level is zero) should often occur at a relatively low level (say 5-6). The foregoing discussion gives a rationale for choosing this algorithm based on the worst case and is not intended as a proof in any sense of the word. The above procedures will enumerate all non-isomorphic cyclic SQS (20). However further tests are needed to show that these designs are in fact nonisomorphic. One technique is to look at the derived triple systems of these designs. Using a procedure initially proposed by Cummings [5] and later developed by Petrenjuk [18], we determined that of the 29 cyclic SQS(20), 25 have nonisomorphic derived triple systems and hence must be nonisomorphic. A full description of this procedure can also be found in Lindner and Rosa [ l S ] or M. Colbourn [4]. For those cyclic SQS (20) that have isomorphic derived triple system. (i.e. I.A.l, I.A.3. and 1II.A. 1, III.A.2, III.B.2, III.B.3) another procedure (described by Gibbons [ 101) is used. We count the number of block triangles in the quadruple system as well as t h e number of such triangles that involve each particular block. A block triangle of a SQS (n),( P , B),is a triple of quadruples b,, bi,bk from B such that any 2 quadruples have 2 points i n common but n o point is common to all 3 blocks (i.e. hi nb, n b k =g). The total number of block triangles is different for each of these cyclic SQS (20) and hence they are nonisomorphic. Both of these procedures usually give the automorphism partitioning of the points of the designs. This will be considered later in our discussion of the results.
nF==,
nf=,
4. Results of tbe enumeration
The main results are that there are 29 nonisomorphic cyclic SQS (20). Of these I S have no short orbits of size 10: 6 have 4 short orbits of size 10 (4 of these
On cyclic Steiner systems S(3,4,20)
287
where originally constructed by Cho [3]). The remaining 8 have 2 short orbits of size 10 with each subcase A, B, C, D having at least one cyclic SQS (20). The only isomorphisms between cyclic SQS (20) are those corresponding to the units of 220. There are 6 cyclic SQS (20) having the unit 13 as an automorphism giving 12 distinct SQS (20); 4 designs have the unit 3 as an automorphism giving 8 distinct SQS(20); 4 designs have the unit 19 as an automorphism giving 16 distinct SQS (20) (one of these was discovered by Griggs and Grannell [ l l ] another by Jain [14]); only one SQS (20) has the unit 9, and no other units as its automorphism giving 4 distinct SQS (20). None of the cyclic SQS (20) has 11 as an automorphism. Hence there are 152 distinct SQS (20) having (Zzo,+) as its automorphism group, but only 29 nonisomorphic cyclic SQS (20). Let us consider the full automorphism group G, for these cyclic SQS (20) and look at the stabilizer of a point (e.g. Go). Then the derived triple system associated with this point, 0, must have Go as a subgroup of its automorphism group. In all but 4 instances the automorphism group of the derived triple system is precisely Go which as it turns out is always a subgroup of the group of units of Zz0. In particular, 14 of these cyclic SQS (20) have derived triple systems that are automorphism-free. For III.B.2, III.B.3, 1II.A.1 and III.A.2, the derived triple systems are all isomorphic and have an automorphism group of order 144 (e.g. Table B.5). In what follows any reference to derived triple systems can be assumed to refer to the derived triple system associated with the point zero. The permutation (2, 12) (4,14) (6,16) (8,18) is an isomorphism between the derived triple systems of III.A.1 and III.A.2, between those of III.B.2 and III.B.3 as well as between those of I.A.1 and I.A.3. An isomorphism from the derived triple systems of III.B.2 to that of III.A.1 is (2, 12) (3,7,5) (13, 17, 15) (4,8,9, 14, 18, 19). The generators of the automorphism group of the derived triple system of III.A.l along with the derived triple system itself is listed in Table B.5. The routine used to generate this automorphism group is based on the work of Druffel er al. [7] and Gibbons [lo]. One further remark; this derived triple systems has a large number of subsystems of order 7. Although the derived triple systems are isomorphic these quadruple systems have different automorphism groups. In two cases III.B.2 and III.B.3, Go, the subgroup of automorphisms fixing zero is precisely the automorphism group of the derived triple system and hence lGol= 144. In the other two cases III.A.1 and III.A.2, Go is merely the unit 13. This is easy to see. Consider III.A.l; the derived triple systems (see Table B.5) has a subgroup of order 8 that fixes 1. None of these are automorphisms of III.A.l. Thus there can be at most one automorphism mapping 1 to j (and fixing 0) for any j > l . Since the unit 13 is an automorphism, there are only 5 possible choices for j . Testing these possibilities out by hand, we see that the unit 13 is the only possible automorphism. Finally, in Table A we list the base blocks and difference quadruples for each of the 29 nonisomorphlc cyclic SQS(20). The block # in this listing refers to the
2 88
K.T.Phelps
Table B.5. Derived triple system of III.A.1 and generators of its automorphism group 1 1 1
3 8 4
4 5 6 7 19 6 3 19 2 3 4 16 15 1. 19 7 6 ’ 15 9 11
2 2 6
1
9
2
1
3
4
7 8 4 3 8 7 8 7 5 8 6 18
9 15 15 2 14 12
10 11 12 10 11 12 10 11 16 10 11 19 10 12 1 1 10 13 14
13 18 14 13 13 7
14 15 16 17 18 19 17 9 16 14 13 5 13 9 12 18 17 5 1 4 6 5 18 17 13 9 17 16 15 18 4 5 1 9 1 1 6 8 2
144
ORDER
PARTITION :
1
1
1
1
7 12 5 13 17 7 9 16 6 9 17 6 8 13 5 12 18
8 9 14 2 4 1 7 0 11 17 1 17 18 6 10 16 2 7 14
13 16 18 1 5 9 1 13 14 1 7 8 1 10 11 4 9 18
6 14 15
4
3 14 2 2 2
8 1 6 18 19 9 1 3 10 12 6 11
1
1
1
1
11 12 16 3 6 1 8 4 11 13 7 11 18 7 10 17 5 10 15 4 1 1 3 3
15 16 6 1 2 15 19 9 1 2 10 13
1 1 0 8 4 3 6 8 9
1
12 15 8 19 11 14 7 1 9 10 18 11 15
2 15 18 3 5 7 16 17 15’ 1 3 4 4 10 14
1
1 12 5 12 2 9 5
1
1
1
1
1
1
14 17 11 19 13 19 3 1 9 10 19 1 4 16
3 15 17 2 5 8 1 2 16 4 5 6 7 13 15
associated edge in the hypergraph (see Table B.3 and B.4). Otherwise the listing should be self explanatory.
5. Problems
There are a host of problems relating to cyclic S Q S ( n ) . For instance are the necessary conditions on n for an S-cyclic SQS (n)to exist, sufficient? Can one find necessary or sufficient conditions for a cyclic SQS ( n ) to have specific additional automorphisms such as particular subgroups of the units of Z,, or automorphisms different from the units of Z,,? Of course, there is the basic problem of determining the spectrum for cyclic SQS (n). Finally, the program was modified t o search for cyclic SQS(22). During a 3 minute run 7 nonisomorphic cyclic SQS(22) where found. Indications are that there will be many (100-200) such systems. However, a complete enumeration will have to wait until the necessary resources are available.*
References [ 11 <’. Berge. Graphs and Hypergraphs, (North Holland, Amsterdam, 1976). [2] R.D. Carmichael. Tactical configurations of rank two, Arner. J. Math. 53 (1931) 217-240. 1.31 C.J. Cho, A study in Steiner quadruple systems, M.Sc. Thesis, McMaster University, Hamilton (1979).
* This problem has now
been solved; see the paper by 1. Diener in this volume
On cyclic Steiner systems S(3,4,20)
289
[4] M. Colbourn, An analysis technique for Steiner triple systems, Proc. 10th S. E. Conf. on Comb., Graph Theory and Computing (1979) (to appear). [5] L.D. Cummings, On a method of comparison for triplesystems, Trans. Amer. Math. SOC.15 (1914) 31 1-327. [6] J. Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1971-72) 315-319. [7] L. Druffel, D. Schmidt, and D. Wang, A generator set for representing all automorphisms of a graph, S.I.A.M. J. Appl. Math. 34 (1978) 593-596. [8] F. Fitting, Zyklishce Lijsungen des Steinerschen Problems, Nieuw Arch. Wisk. 11 (2) (1915) 140-148. [9] M.R. Carey and D.S. Johnson, Computers and Intractability (Freeman, San Francisco, 1979). [lo] P. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. Thesis, University of Toronto (1976). [I 11 T.S. Griggs and M.J. Grannell, Mimeographical notes, Preston Polytechnic, Great Britain. [I21 M. Guregovh and A. Rosa, Using the computer to investigate cyclic Steiner quadruple systems, Mat. Casopis Sloven. Akad. Vied 18 (1968) 229-239. [13] Huppert, Endliche Gruppen I (Springer Verlag, Berlin, 1967). [14] R.K. Jain, On cyclic Steiner quadruple systems, M.Sc. Thesis, McMaster University, Hamilton (1971). [I51 C.C. Lindner and A. Rosa, On the existence of automorphism-free Steiner triple systems, J. Algebra 34 (1975) 430-443. [I61 C.C. Lindner and A. Rosa, Steiner quadruple systems-a survey, Discrete Math. 21 (1978) 147-181. [17] R. Peltesohn, Eine Likung der beiden Heffterschen Differenzenprobleme,Composition Math. 6 (1939) 251-257. [I81 A.Ja. Petrenjuk, Priznaki neizomorfnosti sistem trojek Stejnera, Ukr. Mat. 2. 24 (1972) 772-280, 861; English translation: Tests for nonisomorphic Steiner triple systems, Ukr. Mat. Zh. 24 (1972) 620-626). [ 191 K.T. Phelps, A note on the construction of cyclic quadruple systems, Colloq. Math. (to appear). [20] A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972) 61-71. [21] L. Teierlinck, The existence of reverse Steiner triple systems, Discrete Math..6 (1973) 301-302.
Table A. Table Cyclic SQS (20)
I. No short orbits other than (5,5,5,5)-{1,6, 11, 16}, A. the unit 13 as an additional automorphism.
I.A.I. (see Phelps [19]) Block # (65) (163) (93) (201) (105) (85) (14) (37) (102) (210) (43) (204) (159) 1141)
Difference Quadruples <1,8,1,10> <3,4,3,10> <2,2,8,8’ <4,4,6,6> <2,4,2,12> <2,1,6,11> <1,2,5,12> <1,4,9,6> <2,3,8,7> <5,1,3,11> <1,5,7,7> <4.5,2,9> <3,3,9,5> ~3,1,1,15>
Base Blocks {1,2,lO,lll t1,4,8,111 {1,3,5,13} t1,5,9,151 {1,3,7,9} {1,3,4,101 {1.2,4,91 I1.2,6,15) (1.3,6,141 [1.6,7,10} {1,2,7,14} {1,5,10,12) {1,4,7,161 {1,4,5,61
290
K.T. Phelps
LA2 Block #
Difference Quadruples
BaSe Blocks
<1,8,1,10> <3,4,3,10> <2,2,8,8> <4.4,6,6> <2,4,2,12> <2,1,6,11> <1,3,1,15> <1,7,5,7> <4.5,4,7>
<3,5,3,9> <1,2,3,14> <5,1,1,13> <2,5,6,7> <3,2,9,6>
bA.3 Difference Quadruples
Base Blocks
(65)
<1,8,1,10>
t1,2,10,111
(163) (93)
<3,4,3,10>
{1,4,8,111
<2,2,8,8j
(1,3,5,131
(201)
<4,4,6,6>
(1,5,9,151
Block #
On cyclic Steiner systems S(3,4,20)
B. The unit g as an additional automorphism. Difference Quadruples
Block #
Base Blocks
C. The unit 19 as an additional automorphism
I.C.l. S-cyclic (see Jain [14]) Difference Quadruples
Block # ~
~~
<1,1,9,9> <2,2,8,8> 0,3,7,7> <4,4,6,6> <2,4.2,12> <1,3,1,15> <1,5,1,13> <2,1,2,15> <2,5,2,11> <3,4,9,4> <3,5,7,5> <3,6,3,8> <4,5,6,5>
Base Blocks
29 1
K.T. Phelps
292 I.C.2.
I.C.3.
Block #
Difference Quadruples
Base Blocks
<1,1,9,9> <2,2,8,8> <3,3,7.7> <4,4,6,6> <1,2,4,13> <1,3,9,7> <1,5,3,11> <1,7,9,3> <2,3,2,13> <2,6,5,7> <2,7,5,6> <3,5,1,11> <4,1,4,11> <4,2,1,13>
(1,2,3,121 (1,3,5,131 {1,4.7,141 (1,5,9,151 (1,2,4,8) (1,2,5,141 {1,2,7,101 {1,2,9,18) (1,3,6,8) {1,3,9,14) (1,3,10,15) (1,4,9,10) {1,5,6,10) 11,5,7,81
O n cyclic Steiner systems S(3,4, 20)
I.C.4. (Originally due to T.J. Griggs and Grannell [ 111) Block #
Difference Quadruples
Base Blocks
D. No unit as an additional autornorphism. All systems are autornorphisrn free.
triple
1.D.1.
Block #
Difference Quadruples
Base Blocks
<1,1,2,16> <1,7,8,4> <1,4,2,13> <2.5,6,7> <4,3,1,12> <3,7.4,6> <3,6,4,7> <2,4,5,9> <1,8,1,10> <2,3,4,11> <2,6,2,10> <3,2,1,14> <3,5,3,9> <1,5,8,6’
!1,2,3,51 {1,2,9,171 {l,2,6,81 {1,3,8,14) {l,5,8,9) {1,4,11,151 { l r 4 ,10,141 {1,3,7,121 {1,2,10,111 {1,3,6,101 {1,3,9,111 {1,4,6,71 {1,4,9,121 !1,2,7,15}
Difference Quadruples <1,1,2,16> <1,7,8,4> <1,4,4,11> <2,4,6,8> <1,5,1,13> <1,9,7,3> <2,7.1,10> <6,2,1,11> <2,3,6,9> <2,5,9,4> <2,6,7,5> <4,3,2,11> <3,3,4,10> <3,5,3,9>
Base Blocks [1,2,3,51 {1,2,9,171 {1,2,6,101 {1,3,7,131 {1,2,7,81 {1,2,11,181 {1,3,10,111 {1,6,9,101 {1,3,6,121 11,3,8,14 1 {1,3,9,161 {1,5,8,101 {1,4,7,111 [1,4,9,12?
1.D.2.
Block # (1) (62) (33) (109) (38) (78) (130) (216) (100) (121) (127) (196) (154) (172)
293
K.T. Phelps
294
I.D.3.
Block # (88)
(111) (17) (151) (69) (157) (41) (5) (211) (20) (132) (194) 1165) (170)
Base Blocks
Difference Ouadruples
<2,2,3,13-. <2,4.8,6> <1,2,8,9’ 3,2,9,6\ <1,8,5,6> <3,3.7,7’ ~1,5,4,10> <1,1,6,12> <5,2,1,12> <1,3,1,15> <2,7,4,72 <4.2,10,4, < 3,4,5,8. < 3,5,1,11.*
11.3,5,81 11,3,7,15) (1,2,4,12) (1,4,6,15) {1,2,10,15) (1,4,7,14) (1,2,7,11) (1.2,3,9? (1.6,8,9) 11.2,5.6) (1.3.10.14) {1,5,7.17) 11,4,8,13) (1.4.9.10) ~-
I.D.4. -
~~
I4loc.k #
Difference Ouadruples
<1,2,2,15> <2,3,1,14> <1,3,3,13’ <4,4,1,11> <6,1,1,12, (3.4.6,7’ <1,8.2,9> <1,7,10,2, ‘. 1,5,4,10, <2,6,8,4’ .: 2,7,4,7’ ~2.5,8,5> <3,5,3,9> 3,6,5,6,
Base Blocks
(1,2,4,6) {1,3,6,7) (1,2,5,8) {1.5,9,10) (1,7,8,9) (1.4,8,14) (1,2,10,12) (1,2,9,19? (1,2,7,11) {1.3,9,17) {1,3,10,14 i (1,3,8,16) (1,4,9,12} {1,4,10,15)
I.D.5. Block #
Difference Ouadruples
1,2,2,15’ 2, 3 , l . 14 3.3,7,7 2,5,2,11, 1 , 1,10,8. 1,6,9,4’ 3,6.6.5 ,1.5.4.10 2,6,3,9’ ‘ 1,3,5,11 1,7.5,7 .4,2,1,13’ -2,8.4,6’ 3,4,4,9’
Base Blocks
!1,2,4,63 11,3,6,7! i1,4,7,14) i1,3,8,10) (1,2,3,131 (1,2,8,17) i 1,4,10,16I 11,2,7,11\ t 1,3,9,12! ~1,2,5,101 (1,2,9,14) (1,5,7,81 {1,3,11,15) :1,4,8,12)
On cyclic Steiner systems S(3,4,20)
I.n.6. ~
~~~
~
Difference Quadruples
Block ,#
Base Blocks
<1,2,2,15> <2,3,4,11> <3,3.7,7> <3,5,8,4> <1,3,6,10> <1,7,9,3> <2,6,3,9> <5,2,1,12> <1,1,10,8> <2,4,1,13> <1,5,9,5> <1,6,2,11> <2,8,6,4> <4,4,5,7,
I.D.7. Difference Quadruples
Block #
Base Blocks
<1,2,2,15> <2,3,3,12> <1,7,10,2> <1,1,10,E D <3,6,3.8>
<1,3,5,11> <2,4,8,6> <1,5,1,13> <1,9,6,4> <4,3,1,12> <3.4,6,7> <4,2,9,5> <2,7,4 ‘7> <2,5,8,5>
11. Two short orbits other than (5,5,5,S ) - { l , 6, 11.16).
A. Short orbits:
( 2 . 8 , 2 , 8 ) { l , 3.11, 13), (4.64.6) {1,5,11,15}.
1I.A.1. Unit 13 as additional automorphism Block #
Difference Quadruples <3,5,8,4> <3,3,7,7> <1,1,9,9-. <4,2,3,11’ <2,4,1,13’ <2,1,6,11> <5,1,2,12> <2,6,5,7> <1,4,4,11’ <1,7.9,3> <1,3,9,7> <3,2,2,13’
Base Blocks I1,4.9,171 {1,4,7,141 I1,2,3,12} {1,5,7,101 I1,3.7,8} (1,3,4,10) {5,1,2,121 !1,3,9,14} {1,2,6,101 !1,2,9,18; {l, 5,6,8? {1,3,9,7? I1,4,6,8}
295
K.T. Phelps
296
ll.A.2. Automorphism free derived triple system Block #
Difference Quadruples
Base Blocks
<1,1,2,16> < 1,6,3,10> < 3,6,1,10> < 2,6,8,4> <’ 1 ,5,2,12> <3,2,1,14> <2,3,4,11> <2.5,6,7’ <2.4,5,9> c l , 8,7,4> “3,5,3,9> <1,4,7,8> <4,3,1,12>
B. Short orbits:
( 1 . 9 . 1 . 9 ) {1,2,11,12}, (3.7.3.7) {1,4,11,14}.
U.B Unit 3 as additional automorphism ~~~
~
Block #
(215) (2) (159) (117)
(123)
Difference Quadruples
Base Blocks
<1,8,6,5> <3,2,2,13> <3,5,8,4> <4,4,1,11> ‘. 1,7,9,3> <2,1,6,11> <6,1,2,11> < 1,1,3,15> <3,3,9,5> <2,5,4,9> <1,5,7,7> <2,4,10,4> <2,6,2,10>
C. Short orhits: (1 9. 1 . Y.)
I1,2,7,141 ll, 3,7,171 !1,3,9,11}
( 2 . 8 , 2,s).
n.C.1. Automorphism free derived triple system
Block #
Difference Quadruples ,I, 2 , L , i s ,. <3,4.3,10’ <2,4,10,4> <2,3,6,9> <3,3,5,9’ <5,2,1,12> c5,1,3,11> ~1,8,8,3’ <4,4,1,11‘ < l ,1,6,12> <1,5,8,6> <2,5,7,6> <2,7,4.7’
Base Blocks t1,2,4,6r {1,4,8,111 {1,3,7,171 (1,3,6,12} {1,4,7,121 {1,6,8,91 (1,6,7,10} {1,2,10,18} [1,5,9,101 {1.2,%9) [1,2,7.15) (1,3,8,151 (1.3.10.14)
On cyclic Steiner systems S(3,4,20)
II.C.2. Automorphism free derived triple system
Block #
Difference Quadruples
Base Blocks
<1,2,2,152 <3,7,4,6> <3,6,4,7, <2.4,5,9> <2,1,5,12’ <4,1,1,14> <2,3,4,11, <4,3,1,12> <1,3,8,8> c1,6,2,11> < 1 ,7,6,6’ <2,5,8,55 <3,3,9,5>
D . Short orbits:
(1.9, 1.9) {1,2,11,12} (4,6,4,6)11.5, 11, lS}.
Il.D.1. Derived triple system is automorphism free Block #
Difference Quadruples
Base Blocks
<1,1,2 <2,3,7 <2,8,7 <3,3,9 <6.2,1 <2,6,8 <1,4,4 <3,4,1 <1,5,1
II.D.2. Automorphism free derived triple system Block ,#
(116)
Difference Quadruples < l ,1,2,16> <3,5,2,10> <2,5,3,10> <3,3,1,13> <1,8,9,2> <4,2,3,11> <2,6,5,7> <2,7,5,6> <1,4,1,14> <1,6,6,7> <1,7,4,8> <3,2,4,11> <4,3,9,4>
Base Blocks
297
K.T. PheJps
298
II.D.3. Automorphism free derived triple system Block #
Difference OuadruDies
Base Blocks
<1,2,2,15> <2,6,2,10> <3,4,3,10> <2,1,6,11> <5,1,1,13> <4,2.3,11> <2,4,7,7> <1,5,6,8> <2.5,4,9> <4,1,3,12> <1,7,5,7> <1,8,8,3> 3,3,5,9> 111. Four short orbits plus ( 5 5 . 5 , s ) - (1.6. 11, 16)
(1,9,1.9) (2,8,2,8) ( 3 . 7 . 3. 7) (4.6.4.6)
{1.2,11,12}, (1.3,11,13}. (1.4. 1 1 . 14). {l,S,ll.l5].
A. The unit 13 as an additional automorphism.
IILA.l. (see Cho [ 3 ] for original construction) Block # (33) (69) (42) (148) (10) (46)
(132) (179) (98) (186) (119) (108)
Difference Quadruples
Base Blocks
<1,4,4,11> <3,5,8,4> <1,5,6,8> * 3,2,2,13> <1,2,1,16> <1,6,1,12> <2,7,4,7> ‘3,6,3,8’ <2,3,3,12, <4,1,1,14> <2,5,7,6> .2,4,5,9>
(1,2,6.10) (1.4.9,17} (1,2,7,131
(1,4,6,8} {1,2,4,5) [1,2,8.9) !1,3,10,14) {1,4,10,13) t1,3,6.9) i1,5,6,7i (1,3,8,15) 11,3,7,12)
IIl.A.2. (see Cho [ 3 ] ) Difference Quadruples
Base Blocks
<3,5,8,4, <1,5,6,8> <3,2,2,13> <2,3,3,12> <4,1,1,14. <2,5,7,6, <2,4,5,9> <1,3,9,7> <1,7,9,3> <2,1,6,11> <6,1,2,11>
[1,2,6,10) [1,4,9,17) 11,2.7.13) {1,4.6.8) {1.3,6,9) !1,5,6,71
(1,3,8,15) {1.3.7,12) {1,2.5,14 1 (1,2,9,18) {1,3,4,10) (1,7,8,10)
On cyclic Steiner systems S ( 3 , 4 , 2 0 )
B. The unit 3 as an additional autornorphism.
UI.B.l. Block #
(159) (117) (43)
Difference Quadruples <1,8,6,5> <3,2,2,13> ~3,5,8,4> <4,4,1,11> <1,7,9,3> <2,1,6,11> <6,1,2,11> <2,4,2,12> <1,1,3,15> <3,3,9,5> <2,5,4,9> <1,5,7,7>
Base Blocks {1,2,10,161 {1,4,6,81 t1,4,9,171 {1,5,9,101 {1,2,9,181 {1,3,4,101 {1,7,8,101 {lr3,7,91 {lr2,3,61 {1,4,7,161 {1,3,8,121 {1,2,7,14 1
III.B.2. (see Cho [3]) Block #
Difference Quadruples
Base Blocks
IU.B.3. (see Cho [3])
Block #
Difference Quadruples
148) 69) 176) 199) 3) 98) 193) (119) (28) (631 (85) (215)
<3,2,2,13> <1,8,6,5> <3,5,8,4> <4,4,1,11> <1,1,4,14> <2,3,3,12> <4,2,9,5> <2,5,7,6> <1,3,9,7> <1,7,9,3> <2,1,6,11> <6,1,2,11>
Base Blocks
299
K.T. Phelps
300
IILC. Derived triple system is automorphisrn free
Block #
Difference Ouadruples
Base Blocks
Annals of Discrete Mathematics 7 (1980) 301-313 @ North-Holland Publishing Company.
O N CYCLIC STEINER SYSTEMS S(3,4,22) Immo DIENER Lehrstiihle fur Numerische und Angewandte Mathematik, Uniwrsitat Gattingen, Gomngen, West Germany A Steiner quadruple system of order u is said to be cyclic if it admits an automorphism of order u. In this paper we enumerate all cyclic Steiner quadruple systems of order 22 and establish that there are exactly 21 nonisomorphic cyclic SQS(22) yielding a total of 210 distinct cyclic quadruple systems of order 22.
1. Introduction A Steiner system S(t, k , u ) is a pair ( X , Q ) where X is a u-set and Q is a collection of k-subsets of X such that every t-subset of X is contained in exactly one member of Q. The elements of X are called points and the elements of Q are called blocks. In this paper we are concerned only with the case f = 3 and k = 4. Such a system S(3,4, u ) is also called a Steiner quadruple system of order u (briefly SQS(u)). A Steiner quadruple system of order u is said t o be cyclic if it admits an automorphism of order u. Without loss of generality we assume X=Z, the set of residues mod u and +) as a subgroup of the automorphism group of a cyclic SQS(u). A Steiner quadruple system of order u exists for all u = 2 , 4 mod 6 (Hanani [S]) whereas for cyclic quadruple systems this question is still unresolved. Using a computer Guregova and Rosa [7] have enumerated all cyclic SQS(u) for u S 16. They have found that no cyclic quadruple system exists for u = 8,14,16 and exactly one exists for u = 4 , 1 0 . Others have established the existence of cyclic SQS(u) for various orders u [5, 111. Until recently [13] all cyclic SQS(u) were S-cyclic. Jain [9] constructed the unique S-cyclic SQS(20). Later Phelps [13] as well as Griggs and Grannel [6] have constructed further examples of cyclic SQS(20). Finally Phelps [12] enumerated all cyclic SQS(20). He found that there are exactly 29 nonisomorphic such systems. He further established that all isomorphisms between cyclic SQS(20) are induced by units in the ring of integers mod 20 and that there is a total of 152 distinct cyclic SQS(20). In this paper we consider the next case u = 2 2 and completely enumerate all cyclic SQS(22). There are exactly 21 nonisomorphic such systems yielding a total of 210 distinct cyclic SQS(22). As in the case u = 20 the only isomorphisms between any of these systems are induced by units in the ring of integers mod 22. None of the systems has a unit as an additional automorphism.
(a,
301
I . Diener
302
The author was inspired to consider this order by a result he had obtained in his Diplomarbeit [ 3 ] (see also [4]):
Theorem 1.1. A necessary condition for an S-cyclic SQS(u) to exist is u = 2,4, 10 or 20 mod 24 and u f 0 mod 7. Part of this theorem has independently been found by Phelps [12]. For the definition of S-cyclic cf. [ l 1, 121. At first the aim was to search for a cyclic SQS(u) which is not S-cyclic. The smallest order that seemed promising was u = 22 because such a system must contain at least one orbit of length $u and no quadruple system with this property had been found before. The systems were found by a series of computer runs on a Univac 1108 (Gesellschaft f u r wissenschaftliche Datenverarbeitung (GWD) mbH, Gottingen) and a PDP-15 (Lehrstuhle fur Numerische und Angewandte Mathematik der Universitat Gottingen). In the next sections the methods used are discussed and the results obtained are presented. For further information on quadruple systems the reader is referred to the survey article by Lindner and Rosa [l 13. For undefined terms see also Phelps [121. 2. Some preliminary.considerations
Let r be a unit in the ring of integers mod u. It induces a permutation of the form x -+ rx mod u on the elements of Z,. If we have a cyclic S(t, k , u ) then a unit of Z , will either induce an additional automorphism of the S(t, k , u ) or it will give a distinct (isomorphic) S ( r , k , u ) . I n what follows we shall see how this simple observation greatly reduces the amount of work necessary for the enumeration of cyclic SQS(22). An SQS(22) must contain 385 blocks. Since 385 = 1 1 mod 22 a cyclic SQS(22) must contain an odd number of orbits of size t u = 11. The difference quadruples of all possible orbits with size 11 are A = ( l , 10. 1, lo),
B =(2,9,2,9),
D
E
= (4.7,3,7).
C=(3,8,3,8),
=(5,6,5,6).
The group of units in the ring of integers mod 22 is cyclic of order 10 and the unit 7 is a generator of this group. It induces the following permutation on the set of short orbits: (ADEBC). The induced permutations on the 3-subsets of {A, B,C,D, E ) are ( { A ,B,Cl, { D , C, A ) , { E , A. D ) , {B,D. E ) . { C , E, B))
On cyclic Steiner systems S(3,4,22)
303
Observe that the group of units acts transitively on the set of short orbits and that the 3-subsets of {A,B,C, D, E } decompose into two classes. So, in the enumeration, if we look for systems including exactly one short orbit we can assume that A is included and no other short orbit. Similarly if we look for systems including exactly three short orbits we can assume that either A, B and C or A, B and D are included. Now let us consider the unit -1 =21 mod 22. In a cyclic SQS(22) there must be a block containing the points 0, 1 and 2. The difference quadruples of the possible orbits leading t o blocks that contain these points are
Dn:=(1,1,n+1,22-n-3),
n = l , 2 , . . . , 17
and -1 induces the following permutations on these orbits:
D,
-+DI8-,,
n = 1 , 2 , . . . , 17.
We have already seen that we may without loss of generality include the orbit A into any of the systems to be found. But then D8, D9 and D,, cannot be included because some of the corresponding blocks would intersect in more than two points with blocks corresponding to the orbit A. Since every short orbit is fixed under the permutation induced by the unit -1 we may, in addition to the blocks already mentioned, include exactly one of the orbits D, for n = 1 . 2 , . . . , 7 into the system. Summarizing we have found that the search for a maximal set of painvise nonisomorphic cyclic SQS(22) has to cover only the following cases: 1.n 1Ia.n I1b.n 1II.n
include include include include
A and D,,, A, €3, C and D,, A, B, D and D,,
A, B, C, D, E and D,
( n = l , 2 , . . . ,7).
Any cyclic SQS(22) must be isomorphic to one of the systems included in the Furthermore if above 28 cases, the isomorphism being induced by a unit of Z2*. any of the systems covered by the cases 1-111 are isomorphic, the isomorphism cannot be induced by a unit.
3. The enumeration The enumeration consisted in a series of computer runs. The programs were written in FORTRAN and used simple backtrack search on the possible orbits. However, the first runs were a great disappointment since the program took up far too much computing time. Various improvements had to be made and further knowledge on the system had to be utilized until a reasonable runtime resulted.
30.1
1. Diener
The cases 111.1-111.7 were covered by a Univac 1108 in a total runtime of 1.5 hours. Cases 1.n and 1I.n were investigated on a PDP-15. This machine is much slower than the Univac and because the storage space available was limited to about 20 k 18-bit words further improvements on the program had to be made. The total runtime for the cases 1.1-1.7 was 25 hours and 4 hours for the cases 11.1-11.7. No system was found in case 111. Two systems were found in case I1 and 19 in case I. Since none of these systems is fixed by a unit of Z,, and there are (p(22)= 1 0 units we have a total of 210 distinct cyclic SQS(22). As was pointed out above, if any of the 21 systems found were isomorphic the isomorphism cannot be induced by a unit of Z,,. However, further tests were needed to establish that these systems are in fact nonisomorphic. A program was written to count t h e number of triangles in each of the 21 systems. A triangle is a set of three blocks that intersect pairwise in exactly two points but with no point in common t o all three blocks. This procedure divided the systems into 13 classes, one class containing three systems and six classes containing two systems. Next, we counted the number o f pairs of points that were contained in a fixed number of triangles. This showed that all 21 systems are nonisomorphic. The total runtime for the program that determined the nonisomorphism of all systems was less than one minute on the Univac 1108. In Table I we give a listing of the baseblocks and difference quadruples of the 2 1 nonisomorphic SQS(22) found. Seven of these systems have independently been found by Phelps [12, 141.
4. Some remarks on intersections between cyclic SQS(22) Since any cyclic SQS(22) must include at least one short orbit there cannot be more than five pairwise disjoint cyclic SQS(22). We found that numerous quintuples of pairwise disjoint cyclic SQS(22) exist but essentially only one quintuple with all five systems being isomorphic. We have /72iQ,, n 72JQ,,1= 0 for i, j = 0, 1. . . . , 4 with i# j where Q l r denotes the collection of blocks in system number 12 in Table 1. Note that from this quintuple one can construct a cyclic t-design with A = 14 namely an sl,(3,4,22): Denote by U the set of all 4-subsets of Z,,. Any 3-subset of Z,, occurs in exactly 19 elements of U. Now form the subset
of U. Clearly the elements of W form a cyclic S,4(3, 4,22).
On cyclic Steiner systems S(3.4.22) Table 1
***
SYSTEM N R . 1 B A S E B L O C K S OF A C Y C L I C SQS(22)
22 1
3. 5 )
***
SYSTEY, Na. 3 S A S E S L O C K S OF A C Y C L I C SOS(Z2) (
( ( ( (
1, 1. 1, 1. 1,
2,12r13) 2,
e,201
2,17,19J 3.11.17) 4,14,19)
( 1. 2 . 31 5 )
( ( ( (
1, 1, 1, 1,
2, 9.11) 3, 6,181 3.14.20) 5, 8,15)
9!PFEREY C E QUADRCPLES ( ( ( ( (
1,1n, 1,lC) 1, €,12, 3) 1.15, 2, 4 ) 2, e, e , 6 ) 3,1C, 5 , 4 )
( ( ( ( (
1, 1 , 2 , l F ) 1, 7, 2,121 2, 3.12, 5 ) 2.11, 6 , 3) 4 , 4 , 7 , 7)
305
I . Diener
306
Table 1 (continued)
***
SYSTEF “1.
4
FASIBLOCKS 3P 4 CYCLIC SqS(22) 5)
( 1, 2.12.13
( 1, 2 , 3,
( 1, 2.16.26) ( 1. 3,1@,l7) ( 1, 4, 9.14)
( 1. 3. 5, G ) ( 1. 3.11.141 ( 1, 5,11,15)
DIFFEXEVCL Q U A D 9 U P L l S ( 1.12,
1,lP)
( 1, 6 , l @ , ( 1.14, 4, ( 2, 7 , 7, ( 3. 5 , 5 ,
5)
3) 6) 2)
( ( ( ( (
1, 1, 2,lF) 1. 7 , 2,121 2, 3, 3,141 2 . f. 3. 9 ) 4,
e,
4.
S)
( 1. 2. ?.21)
***
S Y S T T Y H’i.
6
SASESLOCES OF A CYCLIC SCS(22) ( ( ( ( (
1. 1, 1, 1, 1,
2.12.15 2, 8.14 3, e,13 3.11.15 4,12,1.?
( 1. 2 . 5. 5) ( 1 , 2, 3 . 1 a l 1, 3, 7 , S ’ ( 1, 3.12.28)
( 1, 4.14.19)
91 FFEREYCE Q U A 9 3 U P L l S ( i , i @ ,1 , i e )
! 1 , 1 , 2.16)
( 1, 6.11, 4)
( ( ( (
( 2,
5.12, 5 ) ( 2, P. 4, 3 ) ( 3, e, 6 , 3)
1, 7, 1.13) 7, 4. ?.14) 2, 9. 8 , 3 ) 3,1~,,5, 4 )
307
On cyclic Steiner systems S(3,4,22)
***
SYSTEM N R . 9 R A S E B L O C K S OF A C Y C L I C sQS(22) ( ( ( ( (
1, 1, 1, 1, 1,
2,12,13) 2, 8,181 3 , 5,151 3.12.17) 5, 9,161
9IPFERE"IE ( ( ( ( (
( ( ( ( (
1. 1, 1, 1. 1,
2 , 3, 6 ) 2 , 3,1@)
3. 7,191 3,14,2a) 5.13,ia)
( ( ( (
1, 1, 1, 1,
2 , 4,191 2.11.23) 3, E , l l ) 4, 7 , 1 5 )
( 1, 2 , 7,211 ( 1, 2.14.17) ( I, 3,1fl,15)
( ( ( (
1, 1, 7, 3,
2.15, 4 ) 9 , 9, 3 ) 5, 3,121 3, E, 8 )
( 1 , 5,14, 2 ) ( 1.12, 2 , 6 ) ( 2, 7 , 6, 7 ) ( ?, 4 , 5.16)
i 1, 4, a.13)
QUADRUPLES
l , l @ ,1.12) 1, 6.10, 5 ) 2, 2.18, 9) 2, 9, 5. 6 ) 4 , 4 , 7, 7 )
( ( ( (
1, 1 , 3 , 1 7 1 1, 7 , 1.13) 2 , 4.12, 4 ) 2.11. 6 . 3 )
( 4,
e,
5.
5)
308
I. Diener
i 1, 2 , 4 . 2 1 ) i 1, 2 ,1 3 ,2 2 ) 1 , 3,11,13) ( 1, 4, 6,111
( 1 , 2. 5.15) ( 1 , 2,14.19) ( 1 , 3,13,!3) I 1 , 4 , 9.17)
O n cyclic Steiner systems S(3,4,22)
309
Table 1 (continued)
***
SYSTEK VF.
13
SASEBLOCKS OF A CYCLIC S C S ( 2 2 ) ( ( ( ( (
1, 2,12,13) 1, 2 , 7,131 1 , 2, 5,lE) 1, ?,11,14) 1, 4,11,1?)
( 1, 2 , 3. 9 )
( ( ( (
1, 2,10,14) 1, 3, 6,121 1, 3.13.17) 1. 5 ' , 1 0 . 1 7 )
DI FTFRLV C E OGADIUPLES f
l,l@,
( I, ( 2. ( 7, ( 3,
***
E,
1,lZ) 3,
a)
2.13, 5 ) 9, 3?3) 7, 7, C)
S Y S T E Y VTi.
( 1, 1, 5,14) ( 1. a , 4 , 5 ) ( 2, 3. 5.11) 2.10, A , 6 ) ( 4 , 5, 7 , 6 )
( 3, 5,
19
S A S E E L O C K S OF A CYCLIC S G S ( 2 2 ) ( ( ( ( (
1, 2.12.13 1, 2 , 6,11 1 , 3, 5,15 1. 3,12,17 1, 5 , 9.16
( ( ( (
1, 2 , 4.15 1. 2 . 1 4 . 2 8 1. 3, 9.19 1, 4.11.14
5,
3)
I. Diener
310
Iahlr 1 (continued)
***
SYSTFY Y R . 1 E F A S F l L 3 C T S OT A :YVLIL'
( 1. 2 , 1 2 , 1 ? ) ( 1. 2. 7.
a)
(
CCS(22)
1 , 2 , 3.11)
( 1. 2 , 3.15)
311
On cyclic Steiner systems S(3,4,22)
( ( ( (
( ( ( ( (
4,
7 , 4. 7 )
1, 7 , 1.13) 2, 4 , 2 . 1 4 )
3, 4 . e , 7 ) 5 , 5 , 5, 6 )
1. 2, 3. e l 1, 2 , 1 5 , 1 4 ) 1, 3 , 5,111 1, 3,13,16)
( 1, 1 , 2,IE) ( 1, 3.18, 2) ( 2, 7 , 5 , E ) ( 3 , 5, 3.11)
312
I . Diener
Table 2 Order cyclic
c
Existence S-cyclic
N(t)
D(u)
1
1
no
29 21
152 210
yes
25
>
no
21
>
no yes
31
>
Z X
>
32
?
no
no
10 14. 16 20 22 26 28 37 34 3s. 10 11 16
Yes
yes
SO
8
52 56 58 6’. 61 6X
7(I 71 76
no
no
yes yes ye5
Yes
I
YC,. >
no
>
no
ye\ ye,.
ye\ Yes no yes no
yes
,
ye\ 7
ye\
2 1
%?
31
>
> >
110
ye\
?
ycs
> Ch ?
3 1
no
87 86.88 97
?
,725
32.45
80
W.9X 1011
--
25
31
)
no no yes
)
[71 111, [71 r71 ~91.[121 [3], this paper ~51.r71 T. 5.2. 131
110
yc\
>
Reference
no no )
21
32l‘’l’
[31 [31 [7]. T. 5.1
Let me mention another interesting fact concerning the two systems with 19 base blocks : ( 7 ‘ 0 2n , 7’02,,1= 22
for i, j = 0, 1,2, . . . , 9 .
Any one of the systems 7‘Q,, has exactly two short orbits in common with any one of the systems 7 ’ 0 2 , ,and these are the only common blocks. ’The largest intersection is 23 1 blocks. It occurs only between the systems 7’Q,, and -7‘& ( 3 short and 9 full orbits). The largest intersection among the systems with exactly one short orbit is 187 and occurs only between the systems 7 ‘ 0 , and 7 ‘ 0 ,1. 5. Spectrum for small orders
In Table 2 we list the known spectrum for cyclic and S-cyclic quadruple systems of small orders. In this table reference is made t o the following theorems recently obtained by Cho and Phelps:
On cyclic Steiner systems S(3,4,22)
313
Theorem 5.1 (Cho [2J). For u = 2 or 10 mod 12, i f there exists a cyclic SQS(u), then there exists a cyclic SQS(2u). Theorem 5.2 (Phelps [14]). If there exists a cyclic SQS(q + l), where q is a prime power, then there exists a cyclic SQS(qk + 1) for all k > 0. N ( u ) denotes the number of nonisomorphic cyclic SQS(u) and D(u)denotes the number of distinct cyclic SQS(u).
References [I] J.A. Barrau, On the combinatory problem of Steiner, Kon. Akad. Wetensch. Amst. Proc. Sect. 11 (1908) 352-360. [2] C.J. Cho, A study in Steiner quadruple systems, M. Sc. Thesis, McMaster University, Hamilton (1979). [3] 1. Diener, S-zyklische Steinersysteme, Diplomarbeit, NAM der Universitat Gottingen (1979). [4] I. Diener, On S-cyclic Steiner systems, in preparation. [S] F. Fitting, Zyklische Lasungen des Steiner'schen Problems, Nieuw. Arch. Wisk. 11 (2) (1915) 140-148. [6] T.S. Griggs and M.J. Grannell, Mimeographical notes, Preston Polytechnic, Great Britain. [7] M. Guregovl and A. Rosa, Using the computer to investigate cyclic Steiner quadruple systems, Mat. casopsis SAV 18 (1968)229-239. [8] H. Hanani. On quadruple systems, Canad. J. Math. 12 (1960) 145-157. [9] R.K. Jain, On cyclic Steiner quadruple systems, M.Sc. Thesis, McMaster University, Hamilton ( 1971). [ 101 E. Kohler. Numerische Existenzkriterien in der Kombinatorik, in: Numerische Methoden bei Graphentheoretischen und Kornbinatorischen Problemen (Birkhauser, Basel, 1975) 99-108, and E. Kohler, Zyklische Quadrupelsysteme, Abh. Math. Sem. Hamburg, Bd. XLVII (1979) 1-24. [ l l ] C.C. Lindner and A. Rosa, Steiner quadruple systems-a survey, D i m . Math. 22 (1978) 147-181. [ 121 K.T. Phelps, On cyclic Steiner systems S(3,4,20), Ann. D i m . Math. 7. [13] K.T. Phelps, A note on the construction of cyclic quadruple systems, Colloq. Math. (to appear). [14] K.T. Phelps, Infinite classes of cyclic Steiner quadruple systems, Preprint (1979).
This Page Intentionally Left Blank
PART VI
BIBLIOGRAPHY AND SURVEY OF STEINER SYSTEMS
This Page Intentionally Left Blank
Annals of Discrete Mathematics 7 (1980) 317-349 @ North-Holland Publishing Company
AN UPDATED BIBLIOGRAPHY AND SURVEY OF STEINER SYSTEMS Jean DOYEN Uniuersite‘ Libre de Bruxelles
Alexander ROSA McMasrer Uniuenity
Two earlier versions of our bibliography on Steiner systems [D44], [D44.1] listed 400 and 675 titles, respectively. The number of papers concerning Steiner systems continues to grow, and the present listing contains already some 730 titles; all additions are recent papers. As for inclusion into our bibliography of papers dealing with finite projective, affine or inversive spaces, we follow the same policy as in [D44] and [D44.1] Information about reviews in referative journals is attached to the titles in the bibliography; MR stands for Mathematical Reviews while Z stands for Zentralblatt fur Mathematik und ihre Grenzgebiete. 1. A bibliography of Steiner systems W. Ahrens, Mathematische Unterhaltungen und Spiele (B.G. Teubner, Leipzig, 1901; 2. Aufl., B.G. Teubner, Leipzig 1918) 2. Band, Kap. XIV: Anordnungsprobleme. V.E. Aleksejev, 0 Skolemovskom metode pestrojenija ciklifeskich sistem trojek Stcjnera, Mat. Zametki 2 (1967) 145-156. MR 35 #5341; Z 179, p. 28, (English translation: Skolem method of constructing cyclic Steiner triple systems, Math. Notes 2 (1967) 571-576 (1968).) V.E. Aleksejev, 0 fisle sistem trojek Stejnera, Mat. Zametki 15 (1974) 767-774. MR 50 #12754; 2 291.05006. (English translation: On the number of Steiner triple systems, Math. Notes 15 (1974) 461-464.) 1.s.O. Aliev, Simmetrifeskije algebry i sistemy Stejnera, Dokl. Akad. Nauk SSSR 174 (1967) 511-513. MR 35 #2756; 2 213, p. 291. (English translation: Symmetric algebras and Steincr systems, Soviet Math. Dokl. 8 (1967) 651-653.) I . S . 0 . Aliev, Kombinatornyje schemy i algebry, Sib. Mat. 2. 13 (1972) 499-509. MR 46 # 5 140; Z 285,05025. (English translation: Combinatorial designs and algebras, Sib. Math. J. 13 (1972) 341-348.) 1.S.O. Aliev and E. Seiden, Steiner triple systems and strongly regular graphs, J. Combinatorial Theory 6 (1969) 33-39. MR 38 #4362; 2 175, p. 503. I.S.0. Aliev and M.F. Semionov, Process obmena dlja sistem fetviorok Stejnera, Sbornik statej PO matematike, Jakutsk 1975, p. 43. 1.S.O. Aliev and F.I. Trofimceva, Process obmena i sistemy fetviorok Stejnera, Sbornik statej PO matematike, Jakutsk (1975) 44-45. W.O. Alltop, Extending r-designs, J. Combinatorial Theory (A) 18 (1975) 177-186. MR 5 1 #10131; Z 297.05028. S.L. Alpert, Two-fold triple systems and graph imbeddings, J. Combinatorial Theory (A) 18 (1975) 101-107. MR 51 #185; 2 299.05015. 3 17
3 18
J . Doyen, A. Rosa
[A10.1] B. Alspach and B.N. Varma, On almost Steiner triple systems, Proc. Tenth S.-E. Conf. Combinatorics. Graph Theory and Computing, Boca Raton, 1979 (to appear). [A10.2] L.D. Andersen. A.J.W. Hilton, and E. Mendelsohn, Embedding partial Steiner triple systems (to appear). 1. Anderson, A First Course in Combinatorial Mathematics (Clarendon Press, Oxford, 1974) Chap. 7: Steiner systems and sphere packings, pp. 97-116. MR 49 #2402; Z 268.05001. R.R. Anstice, On a problem in combinations, Cambr. and Dublin Math. J. 4 (1852) 279-292. R.R. Anstice. On a problem in combinations, Cambr. and Dublin Math. J. 4 (1853) 149-154. E.F. Assmus Jr. and M.T. Hermoso. Non-existence of Steiner systems of type S(d - 1, d, 2d), Math. Z. 138 (1974) 171-172. MR 50 #4330; Z 273.50015, Z 282.50024. EX. Assmus Jr. and H.F. Mattson Jr.. Disjoint Steiner systems associated with the Mathieu groups, Bull. Amer. Math. SOC.72 (1966) 843-845. MR 34 #74; 2 158, p. 14. . E.F. Assmus Jr. and H.F. Mattson Jr.. On the number of inequivalent Steiner triple systems. J. Combinatorial Theory 1 (1966) 301-305. MR 34 #4168; Z 158. p. 15. Errata: J. Combinatorial Theory 2 (1967) 394. MR 35 #4116. E.F. Assmus Jr. and H.F. Mattson Jr.. Perfect codes and the Mathieu groups, Arch. Math. 17 (1966) 121-135. MR 34 #4050: 2 144, p. 262. E.F. Assmus Jr. and H.F. Mattson Jr., Steiner systems and perfect codes, University of North Carolina Institute of Statistics Mimeo Series No. 484-1 (1966). E.F. Assmus Jr. and H.F. Mattson Jr.. On tactical configurations and error-correcting codes, J. Combinatorial Theory 2 (1967) 243-257. MR 36 #64; 2 189, p. 191. E.F. Assmus Jr. and H.F. Mattson Jr.. Research problem, J. Combinatorial Theory 3 (1967) 307. E.F. Assmus Jr. and H.F. Mattson Jr., New 5-designs, J. Combinatorial Theory 6 (1969) 122-151. MR 42 #7528; Z 179, p. 29. E.F. Assmus Jr. and H.F. Mattson Jr.. On the automorphism groups of Paley-Hadamard matrices, Combinatorial Mathematics and its Applications, Proc. Conf., Univ. North Carolina. Chapel Hill, NC, 1967 (Univ. North Carolina Press. Chapel Hill, NC, 1969) 98-103. MR 40 #4139: Z 207, p. 26. E.F. Assmus Jr. and H.F. Mattson Jr., Coding and combinatorics, SlAM Review 16 (1974) 349-388. MR 50 #12432; Z 268.94002; Z 286.94009. E.F. Assmus Jr.. H.F. Mattson Jr.. and M. Guza, Self-orthogonal Steiner systems and projective planes. Math. 2. 138 (1974) 89-96. MR 5 0 #12764; 2 273.50016; Z 283.50018. T. Atsumi. On Steiner systems. Rep. Fac. Sci. Kagoshima Univ. No. 6 (1973) 7-10. MR S O #433 1. R.D. Baker. Factorization of graphs, Ph.D. Thesis, Ohio State University, 1975. R.D. Baker, Partitioning the planes of AG,,(2) into 2- designs, Discrete Math, 15 (1976) 205-211. MR 54 #4999; Z 326.05013. R.D. Baker, Quasigroups and tactical systems, Aequat. Math. 18 (1978) 296-303. R.D. Baker and R.M. Wilson, Nearly Kirkman triple systems, Utilitas Math. 11 (1977) 289-296. MR 56 #133; Z 362.05030. W.W.R. Ball, Mathematical Recreations and Problems. 1st. ed., London 1892 (French translation by J. Fitz-Patrick, Hermann, Paris 1898: Rtcrtations et Problemes Mathematiques.) 4th ed. Mathematical recreations and essays, London 1905 (French translation by J. Fitz-Patrick, Hermann, Paris 1908:RtcrCations Mathtmatiques et Problkmes des Temps Anciens et Modernes): 11th ed. (revised by H.S.M. Coxeter) (MacMillan. London, 1939) Chapter 10: Kirkman’s school-girls problem. 2 22, p. 1. W.W.R. Ball, Proposed problem. Educational Times (February 1, 1911) 82. W.W.R. Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, 12th ed. (University of Toronto Press, Toronto, 1974) Chap. 10: Combinatorial designs (by J.J. Seidel). MR 50 #4229. J.A. Barrau, Over drietalstelsels, in het bijzonder die van dertien elementen, Kon. Akad. Wetensch. Amst. Verslag Wis- en Natuurk. Afd. 17 (1908) 274-279 ( = O n triple systems, particularly those of thirteen elements, Kon. Akad. Wetensch. Amst. Proc. Sect. Sci. 11 (1908) 290-295.) J.A. Barrau, Over de combinatorische opgave van Steiner, Kon. Akad. Wetensch. Amst.
Bibliography and suruey of Steiner systems
[B25.1]
[I3261
[B26.1] [B26.2] [B26.3] [B27] [B28] [B29] [B29.1] [B30]
319
Verslag Wis- en Natuurk. Afd. 17 (1908) 318-326 ( = O n the combinatory problem of Steiner, Kon. Akad. Wetensch. Amst. Proc. Sect. Sci. 11 (1908) 352-360). J.A. Barrau, Tripelstelsels van vijftien elementen, Handel. Nederl. Natuur- en Geneesk. Congres 12 (1909) 185-189. S. Bays, Une question de Cayley relative au probltme des triades de Steiner, Enseignement Math. 19 (1917) 57-67. S. Bays, Sur les systtmes de triples de 13 tltments, Enseignement Math. 19 (1917) 332-333. S. Bays, Sur les systtmes cycliques de triples de Steiner, C. R. Acad. Sci. Paris Ser. A 165 (1917) 543-545. S. Bays, Sur les systtmes cycliques de triples de Steiner, C. R. Acad. Sci. Paris Ser. A 171 (1920) 1363-1365. S. Bays, Sur la gtntralisation du probltme des triples de Steiner, Enseignement Math. 22 (1921-22) 66-67. S. Bays, Sur les systtmes cycliques de triples de Steiner, C. R. Acad. Sci. Paris Ser. A 175 (1922) 936-939. S. Bays, Recherche des systtmes cycliques de triples de Steiner difftrents pour N premier (ou puissance de nombre premier) de la forme 6n + 1, J. Math. Pures Appl. (9) 2 (1923) 73-98, S. Bays, Sur les systtmes cycliques de triples de Steiner, Ann. Sci. Ecole Norm. Sup. (3) 40 (1923) 55-96. S. Bays, Sur les systtmes cycliques de triples de Steiner difftrents pour N premier (ou puissance de nombre premier) de la forme 6 n + 1, Ann. Fac. Sci. Univ. Toulouse (3) 17 (1925) 23-61. S. Bays, Sur les systtmes cycliques de triples de Steiner difftrents pour N premier (ou puissance de nombre premier) de la forme 6 n + 1, I. Comment. Math. Helv. 2 (1930) 294-305. 11.411. Comment. Math. Helv. 3 (1931) 22-41; Z 1, p. 264; IV. -V. Comment. Math. Helv. 3 (1931) 122-147. Z 2, p. 182; VI. Comment. Math. Helv. 3 (1931) 307-325. Z 3, p. 100. S. Bays. Sur les systemes cycliques de triples de Steiner differcnts pour N premier de la forme 6n + 1, Comment. Math. Helv. 4 (1932) 183-194. Z 5, p. 49. S. Bays, Sur le nombre de systtmes cycliques de triples difftrents pour chaque classe, Actes Soc. Helvtt. Sci. Nat. 116 (1935) 275-276. S. Bays, Sur les systtmes de caracttristiques appartenant a d = 3, Actes SOC.Helvet. Sci. Nat. 116 (1935) 276-277. S. Bays and G. BelhBte, Sur les systtmes cycliques de triples de Steiner difftrents pour N premier de la forme 6n + 1, Comment Math. Helv. 6 (1933) 28-46. Z 7, p. 195. S. Bays and E. de Weck, Sur les systtmes de quadruples, Comment. Math. Helv. 7 (1935) 222-241. Z 11. D. 194. G.F.M. Beenker, A.M.H. Gerards, and P. Penning, A construction of disjoint Steiner triple systems, TH Report 78-WSK-01, Dept. of Math., Technological Univ. Eindhoven, 4 pp. Z 376.05006. G. BelhBte, Contribution a I’ttude des systtmes cycliques de triples de Steiner difftrents pour N premier de la forme 6n + 1, Thtse de doctorat, Universitk de Fribourg, 1933. G. BelhBte, see [B24] L. Btntteau, Boucles de Moufang commutatives d’exposant 3 et quasi-groupes de Steiner distributifs, C. R. Acad. Sci. Paris Ser. A 281 (1975) 75-76. Z 319.50026. L. Beneteau, Topics about 3-Moufang loops and Hall triple systems (to appear). L. Btntteau, Une classe particulitre de matroides parfaits (to appear). F.E. Bennett, Construction of extended triple systems, Ph.D. Thesis, University of Manitoba, 1976. F.E. Bennett, Extended cyclic triple systems, Discrete .Math. 24 (1978) 139-146. 2 386.05012. F.E. Bennett and N.S. Mendelsohn, On pure cyclic triple systems and semisymmetric quasigroups, Ars Combinatoria 5 (1978) 13-22. F.E. Bennett and N.S. Mendelsohn, On the existence of extended triple systems, Utilitas Math. 14 (1978) 249-267. F.E. Bennett and N.S. Mendelsohn, Some remarks on 2-designs S,(2,3, u) (to appear).
320 [B31]
J . Doyen, A. Rosa
C. Berge. Theorie des Graphes et ses Applications (Dunod, Paris, 1958) MR 21 #1608; Z 88, p. 154. 2nd ed. Dunod, Paris 1963. MR 27 #5246; Z 121, p. 401. (English translation: The Theory of Graphs and its Applications (Wiley, New York, 1962). M R 24A #2381; Z 124. p. 362. Russian translation: Teorija grafov i jejo primenenija (Izd. inostr. lit., Moskva, 1962).) C. Berge. Graph theory, Amer. Math. Monthly 71 (1964) 471-481. MR 30 #3458; Z 121, [B32] p. 401. [B33] E.R. Berlekamp, Coding theory and the Mathieu groups, Inform. Control 18 (1971) 40-64. MR 43 #1734; Z 217. p. 286. [B34] E.R. Berlekamp, J.H. Conway and R.K. Guy. Winning Ways (to appear). Chapter: Turn and turn about. B351 J.-C. Bermond, An application of the solution of Kirkman’s schoolgirl problem: the decomposition of the symmetric oriented complete graph into 3-circuits, Discrete Math. 8 (1974) 301-304. MR 49 #4850; Z 281.05105. [B36] J.-C. Bermond and J. Novhk, On maximal systems of quadruples, Graphs, Hypergraphs and Block Systems, Zielona Cora (1976) 13-19. Z 343.05017. [B37] L. Bertani. Costruzione di spazi di Steiner regolari, Bollet. U.M.I. (4) 11 (1975) 370-374. MR 51 #10120; Z 359.05015. [B381 T. Beth, Algebraische Auflijsungsalgorithmen fur einige unendliche Familien von 3-Designs, Le Mathematiche 29 (1974) 105-135. MR 5 1 #2948; Z 303.05016. [B38.1] T. Beth, On Resolutions of Steiner Systems, Dissertation Erlangen, 1978. V.N. Bhat. On inequivalent balanced incomplete block designs, I., J . Combinatorial Theory 6 (1969) 412-420. MR 42 #1673; Z 169. p. 322. V.M. Bhat, On inequivalent balanced incomplete block designs, II., J. Combinatorial Theory (A) 12 (1972) 260-267. MR 45 #4889. K.N. Bhattacharya, A note on two-fold triple systems, Sankhya 6 (1943) 313-314. MR 5, p. 29: Z 60. p. 313. K.N. Bhattacharya, see [B53] S. Bills, Solution of a problem proposed by W. Lea, Educational Times Reprints 8 (1867) 32-33. S. Bills. see [W24] 1.F. Blake and M. Rahman. A note on generalized Steiner systems, Utilitas Math. 9 (1976) 339-346. MR 53 #10607; Z 332.05014. I.F. Blake and J.J. Stimer, A note on generalized Room squares, Discrete Math. 21 (1978) 89-93. Z 378.05021. 1.F. Blake, see [S51] G.A. Bliss. The scientific work of Eliakim Hastings Moore, Bull. Amer. Math. SOC.40 (1934) 501-514. A.D. Bol’bot. Equationally complete varieties of totally symmetric quasigroups, Algebra i Logika 6 (1967) 13-19. MR 35 #6536. T.R. Booth, Construction of disjoint Steiner triple systems from Steiner quadruple systems, M.Sc. Thesis, Auburn University, August 1976. T.R. Booth, A resolvable quadruple system of order 20. Ars Combinatoria 5 (1978) 121-125. R.C. Bose, On the construction of balanced incomplete block designs, Ann. Eugenics 9 (1939) 353-399. MR 1. p. 199; Z 23, p. 1. R.C. B o x , On a method of constructing Steiner’s triple systems, Contributions to Probability and Statistics, (Stanford Univ. Press, Stanford, CA., 1960) 133-141, MR 22 #11449; Z 94, p. 329. R.C. Bose. On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements, The Golden Jubilee Commemoration Volume (19581959), Part 11. Calcutta Math. Soc. (1963) 341-354. MR 27 #4769; Z 116, p. 112. R.C. Bose and S.S. Shrikhande, On the composition of balanced incomplete block designs, Canad. J. Math. 12 (1960) 177-188. MR 22 #1046; Z 93, p. 319. H.C. Rose, S.S. Shrikhande and K.N. Bhattacharya, On the construction of group divisible incomplete block designs, Ann. Math. Statist. 24 (1953) 167-195. MR 15, p. 3; Z 50, p. 146.
Bibliography and survey of Steiner systems
[~581 rB58.11 [B58.21 [B58.3]
321
R.C. Bose, S.S. Shrikhande and E.T. Parker, Further results on the construction of mutually orthogonal Latin Squares and the falsity of Euler’s conjecture, Canad. J. Math. 12 (1960) 189-203. MR 23A #69; Z 93, p. 319. A. Bray, The fifteen schoolgirls, Knowledge 2 (1882) 80-81. A. Bray, Twenty-one school-girl puzzle, Knowledge 3 (1883) 268. A.E. Brouwer, A note on the covering of all triples on 7 points with Steiner triple systems, Mathematisch Centrum Amsterdam, ZN 63/76, pp. 1-8. A.E. Brouwer, Two new nearly Kirkman triple systems, Utilitas Math. 13 (1978) 311-314. Z 379.05008. A.E. Brouwer, Some non-isomorphic BIBDs B(4, 1; u ) , Mathematisch Centrum Amsterdam, ZW 102/77, 7 pp. Z 366.05011. A.E. Brouwer, Steiner triple systems without forbidden subconfigurations, Mathematisch Centrum Amsterdam, ZW 104/77, 8 pp. Z 367.05011. A.E. Brouwer, Optimal packings of K4’s into a K,, J. Combinatorial Theory (A) 26 (1979) 278-279. R.H. Bruck, What is a loop?, in: Studies in Modern Algebra (A.A. Albert, ed., Prentice Hall, Englewood Cliffs, NJ, 1963) 59-99. Z 199, p. 52. R.H. Bruck, Construction problems in finite projective spaces, Finite Geometric Structures and their Applications (C.I.M.E., I1 Ciclo, Bressanone, 1972) (Edizioni Cremonese, Rome 1973) 105-188. MR 49 #7159; Z 264.05013. R.H. Bruck and H.J. Ryser, The non-existence of certain finite projective planes, Canad. J. Math. 1 (1949) 88-93. MR 10, p. 319; Z 37, p. 375. G. Brunel, Remarques sur les systemes de triades, Mem. SOC.Sci. Phys. Nat. Bordeaux (4) 5 (1894) 47 G. Brunel, Systbmes de n-ades formtes avec n 2 bltments, Procis-verbaux Soc. Sci. Phys. Nat. Bordeaux (1894-95) 56-60. G. Brunel, Sur les systbmes de triades formees avec 6n + 1 tlements, C. R. Assoc. Fr. Av. Sci. 24 (1895). G . Brunel, Sur les triades formtes avec 6n - 1 et 6n - 2 Cltments, Procbs-verbaux SOC.Sci. Phys. Nat. Bordeaux (1895-96) 40-43. G. Brunel, Sur un probleme combinatoire de Steiner, Pro&-verbaux SOC.Sci. Phys. Nat. Bordeaux (1896-97) 37-41. G. Brunel, Sur les deux systemes de triades de treize elements, J. Math. Pures Appl. (5) 7 (1901) 305-330 (or Mem. SOC.Sci. Phys. Nat. Bordeaux (6) 2 (1902) 1-23). F. Buekenhout, Remarques sur I’homogCntitt des espaces lintaires et des systemes de blocs, Math. Z. 104 (1968) 144-146. MR 38 #5647; 2 159, p. 302. F. Buekenhout, Une caracterisation des espaces affins b a k e sur la notion de droite, Math. Z. 111 (1969) 367-371. MR 42 #8383; Z 179, p. 256. F. Buekenhout, A characterization of the affine spaces of order two as 3-designs, Math. 2. 118 (1970) 83-85. MR 42 #7532; Z 202, p. 511. F. Buekenhout, On the orbits of collineation groups, Math. A. 119 (1971) 273-275. MR 43 #7350; Z 206, p. 236. J. van Buggenhaut, On some Hanani’s generalized Steiner systems, Bull. Soc. Math. Belg. 23 (1971) 500-505. MR 47 #8324; Z 254.05013. J. van Buggenhaut, On some non-isomorphic (3,4, u ; 3)-designs, Rend. Mat. (6) 5 (1972) 307-317. MR 50 #9613; Z 241.05019. J. van Buggenhaut, On the existence of 2-designs S,(2,3, u ) without repeated blocks, Discrete Math. 8 (1974) 105-109. MR 48 #8259; Z 276.05021. J. van Buggenhaut, Existence and constructions of 2-designs S,(2,3, u ) without repeated blocks, J. Geometry 4 (1974) 1-10. MR 48 #8260; Z 271.05006. R.J. Bumcrot, Finite hyperbolic spaces, Atti del Convegno di Geometria Combinatoria c sue Applicazioni (Perugia 1970). Universita degli Studi di Perugia (1971) 113-130. MR 49 #6013; Z 226.50019. W. Burnside, On an application of the theory of groups to Kirkman’s problem, Messenger Math. (2) 23 (1893-943 137-143.
1.Doyen. A. Rosa F.C. Bussemaker and J.J.Seidel, Symmetric Hadamard matrices of order 36, Ann. N.Y. Acad. Sci. 175 (1970) 66-79. MR 44 #3894. F.C. Bussemaker and J.J. Seidel, Symmetric Hadamard matrices of order 36, Technological University Eindhoven, Report 70 WSK-02. July 1970, 68 pp. MR 43 #1863. W.H. Bussey. On the tactical problem of Steiner, Bull. Amer. Math. Soc. 16 (1909) 19-22. W.H. Bussey. The tactical problem of Steiner, Amer. Math. Monthly 21 (1974) 3-12. W.H. Bussey. see [V6]. P.J. Cameron, Characterizations of some Steiner systems, parallelisms and biplanes, Math. Z. 136 (1974) 31-39. MR 50 #1908; Z 265.05006; Z 272.05014. P.J. Cameron, Minimal edge-colourings of complete graphs, J. London Math. SOC.(2) 11 (1975, 337-346. MR 54 #10080; Z 312.05107. P.J. Cameron, Two remarks on Steiner systems. Geomctriae Dedicata 4 (1975) 403-418. MR 53 #7805; Z 324.05013. P.J. Cameron, On basis-transitive Steiner systems, J . London Math. SOC.(2) 13 (1976) 393-399. Z 331.05008. P.J. Cameron, Parallelisms of Complete Designs, London Math. SOC.Lecture Note Series No. 23. Cambridge Univ. Press, Cambridge 1976. MR 54 #7269; Z 333.05007. P.J. Cameron and J.H. van Lint, Graph Theory, Coding Theory and Block Designs, Cambridge University Press, London Math. SOC. Lecture Notes Series 19, Cambridge, 1975. MR 53 #10608: Z 314.94008. R.D. Carmichael. Tactical configurations of rank two. Amer. J. Math. 53 (1931) 217-240. Z 1, p. 10. R.D. Carmichael, Note on triple systems. Bull. Amer. Math. SOC.3R (1932) 695-696. Z 5. p. 340. R.D. Carmichael. Introduction to thc 'Ihcor) of Groups of Finite Order (Ginn. Boston. 1937; reprinted by Dover. New York, 1956). Chap. XIV: Tactical configurations, pp. 415-441. E. Carpmael. Some solutions of Kirkman's 15-schoolgirl problem, Proc. London Math. SOC. 12 ( 1 8 8 0 4 1 ) 148-156. A. Cayley. On the triadic arrangements of seven and fifteen things, London, Edinburgh and Dublin Philos Mag. and J. Sci. (3) 37 (1850) 50-53 (Collected Mathematical Papers 1, 481-484). A. Cayley. On a tactical theorem relating to the triads of fifteen things, London, Edinburgh and Dublin Philos. Mag. and J. Sci. (4) 25 (1863) 59-61 (Collected Mathematical Papers V, 95-97). R. Chaffer, M. Eggen. R. St. Andre and D. Smith, Strong finite embeddability for classes of quasigroups. Algebra Univ. 5 (1975) 257-262. MR 52 #10939: Z 323.08007. [C13.1] P.D. Chawathe,-A geometry associated with the Steiner system S(24,8. 5). Geom. Dedicata 7 (1978) 407-413. [ C l J ] Y. Chen, The Steiner systems S(3,6.26), J. Geometry 2 (1972) 7-28. MR 46 #3332; Z 227.50010; Z 231.50019. [c'IJ.l] C.J. Cho. A Study in Steiner Quadruple Systems. M.Sc. Thesis, McMaster University, April 1979. V + 114pp. [Cl5] P.C. Clapham. Steiner triple systems with block-transitive automorphism groups. Discrete Math. 14 (1976) 121-131. MR 53 #12971; Z 323.05012. [C16] W.H. Clatworthy, The subclass of balanced incomplete block designs with r = 11 replications. Rev. Inst. Internat. Statist. 36 (1968) 7-11. MR 37 #2617; Z 165, p. 531. [<'I71 W.H. Clatworthy and R.J. Lewyckyj, Comments on Takeuchi's table of difference sets generating balanced incomplete block designs, Rev. Inst. Internat. Statist. 36 (1968) 12-18. MR 37 #3704; Z 165, p. 531. [CIS] T. Clausen, Uber eine combinatorische Aufgabe. Archiv Math. Phys. (1) 21 (1853) 93-96. [CIX.I] M.J. Colbourn, An analysis technique for Steiner triple systems, Proc. Tenth S.-E. Conf. Combinatorics. Graph Theory and Computing, Boca Raton, 1979 (to appear). LC191 F.N. Cole, The triad systems of thirteen letters, Trans. Amer. Math. SOC.14 (1913) 1-5. [CZO] F.N. Cole, Kirkman parades, Bull. Amer. Math. Soc. 28 (1922) 435-437.
Bibliography and survey of Steiner sysrems
323
F.N. Cole, L.D. Cummings and H.S. White, The complete enumeration of triad systems in 15 elements, Proc. Nat. Acad. Sci. U.S.A. 3 (1917) 197-199. F.N. Cole, see [WlO] [C22] J.H. Conway, A group of order 8, 315, 553, 613, 086, 720, 000, Bull. London Math. SOC.1 (1969) 79-88. MR 40 #1470; Z 186, p. 323. [C23] J.H. Conway, The miracle octad generator (to appear). J.H. Conway, see [B34]. [C24] H.H. Conwell, The three space PG(3.2) and its groups, Ann. Math. 11 (1910), 60-76. D.G. Corneil, see [G8], [GlO]. [C24.1] J. Coupland, On the Construction of Certain Steiner Systems, Ph.D. Thesis, University College of Wales, May 1975. [C25] G.M. Cox, Enumeration and construction of balanced incomplete block configurations, Ann. Math. Statist. 11 (1940) 72-85. MR 1, p. 199. [C26] H.S.M. Coxeter, Twelve points in PG(5,3) with 95040 self-transformations, Proc. Roy. SOC. London Ser. A 247 (1958) 279-293. MR 22 #11044. H.S.M. Coxeter, see [B7]. [C27] D.W. Crowe, The construction of finite regular hyperbolic planes from inversive planes of even order, Colloq. Math. 13 (1965) 247-250. MR 31 #3918; Z 136, p. 153. [C28] D.W. Crowe, Projective and inversive models for finite hyperbolic planes, Michigan Math. J. 13 (1966) 251-255. MR 33 #1790; Z 158, p. 194. [C29] D.W. Crowe, Steiner triple systems, Heawood’s torus coloring, Csaszar’s polyhedron, Room designs, and bridge tournaments, Delta 3 (1972) 27-32. MR 47 #6505; 2 242.05140. [C30] L.D. Cummings, Note on the groups for triple systems, Bull. Amer. Math. SOC.19 (1913) 355-356. [C31] L.D. Cummings, On a method of comparison for triple-systems, Ph.D. Thesis, Bryn Mawr College, 1914. [C32] L.D. Cummings, On a method of comparison for triple-systems, Trans. Amer. Math, SOC.15 (1914) 311-327. [C33] L.D. Cummings, An undervalued Kirkman paper, Bull. Amer. Math. SOC. 24 (1918) 336-33 9. [C34] L.D. Cummings, The trains for the 36 groupless triad systems on 15 elements, Bull. Amer. Math. SOC.25 (1919) 321-324. [C35] L.D. Cummings and H.S. White, Groupless triad systems on fifteen elements, Bull. Amer. Math. SOC.22 (1915-16) 12-16. L.D. Cummings, see [CZl], [WlO]. [C36] R.T. Curtis. A new combinatorial approach to M,,,Math. Proc. Cambridge Philos. SOC.79 (1976) 25-42. MR 53 #3098; Z 321.05018. E.W. Davis, A geometric picturc of the fifteen school-girl problem, Ann. Math. 11 (1896-97) [Dl] 156-157. R. Deherder, Recouvrements. Thkse de doctorat, Universiti Libre de Bruxelles, 1976, 212 [D2] PP. [D2.1] M. Dehon, Designs et hyperplans, J. Combinatorial Theory (A) 23 (1977) 264-274. MR 57 #5781. M. Dehon, Planar Steiner triple systems, J. Geometry 12 (1979) 1-9. Z 369.05006; 2 [D3] 386.0501 1. P. Dembowski, Kombinatorische Eigenschaften endlicher Inzidenzstrukturen, Math. Z. 75 [D4] (1960-61) 256-270. MR 23A #3098; Z 96, p. 149. P. Dembowski, Mobiusebenen gerader Ordnung, Math. Ann. 157 (1964) 179-205. MR 31 [D5] #1607; Z 137, p. 401. P. Dembowski, Finite Geometries (Springer, Berlin 1968) MR 38 #1597; Z 159, p. 500. [D6] P. Dembowski and D.R. Hughes, On finite inversive planes, J. London Math. SOC.40 (1965) [D7] 171-182. MR 30 #2382; Z 137, p. 146. J. Dines and A.D. Keedwell, Latin Squares and their Applications (Academic Press, New [D8] York, 1976) M R 50 #4338; Z 283.05014.
[C21]
1. Doyen, A. Rosa
R.H.F. Denniston, Some packings of projective spaces, Rend. Accad. Naz. Lincei 52 (1972) 3-0. MR 48 #9541. R.H.F. Denniston, Packings of PG(3, q ) , Finite Geometric Structures and their Applications IC.1.M.E.. I1 Ciclo Bressanone 1972) (Edizioni Cremonese, Roma, 1973) 193-199. MR 49 #7160. R.H.F. Denniston, Uniqueness of the inversive plane of order 5, Manuscripta Math. 8 (1973) 11-19. MR 48 #1947; Z 251.05018. R.H.F. Denniston, Uniqueness of the inversive plane of order 7, Manuscripta Math. 8 (1973) 21-26. MR 48 #1948; Z 251.05019. R.H.F. Denniston, Double resolvability of some complete 3-designs, Manuscripta Math. 12 (1974) 105-112. MR 50 #1910: Z 276.05022. R.H.F. Denniston. Some packings with Steiner triple systems, Discrete Math. 9 (1974) 213227. MR 51 #5350; Z 306.05007. R.H.F. Denniston, Sylvester's problem of the 15 school-girls, Discrete Math. 9 (1974) 229-233. MR 5 1 #5322; Z 285.05002. R.H.F. Denniston, Some new 5-designs, Bull. London Math. SOC.8 (1976) 263-267. Z 339.05019, R.H.F. Denniston. Non-existence of the Steiner system S(4, 10.66). Utilitas Math. 13 (1978) 303-309. Z 374.05011. [D17.1] R.H.F. Denniston, Further cases of double resolvability, J. Combinatorial Theory (A) 26 (1979) 298-303. [D17.2] R.H.F. Denniston, Four doubly resolvable complete 3-designs, Ars Combinatoria 7 (1979) 265-272. [D 17.31 R.H.F. Denniston. Three doubly resolvable complete 3-designs, Manuscripta Math. (to appear). [D18] V. D e Pasquale, Sui sistemi ternari di 13 elementi, Rend. R. 1st. Lombard0 Sci. e Lett. (2) 32 (1899) 213-221. [DlU] L.J. Dickey, Construction of absolute and hyperbolic planes from ruled surfaces and ovoids in three-dimensional finite projective geometries, Ph.D. Thesis, University of Wisconsin, 1970. [D20] L.J. Dickey, Using hyperbolic planes to construct block designs, Proc. 25th Summer Meeting Canad. Math. Congress, Thunder Bay (June 1971) 315-336. MR 50 #144. [D20. I] 1. Diener. S-zyklische Steinersysteme, Diplomarbeit, lnstitut fur Numerische und Angewandte Mathematik, Universitat Gottingen, 1979, 42 pp. J.W. DiPaola, Block designs and graph theory, J. Combinatorial Theory 1 (1966) 132-148. MR 33 #5508. J.W. DiPaola, On a restricted class of block designs games, Canad. J. Math. 18 (1966) 225-236. MR 33 #2420; Z 228.05013. J.W. DiPaola. On minimum blocking coalitions in small projective planes games, SlAM J. Appl. Math. 17 (1969) 378-392. MR 40 #1440; Z 191, p. 496. J.W. DiPaola, When is a totally symmetric loop a group?, Amer. Math. Monthly 76 (1969) 249-252. MR 39 #1583; Z 196, p. 40. J.W. DiPaola. Steiner triples and totally non-symmetric loops, Combinat. Structures and their Applications (Proc. Confer. Calgary, 1969) (Gordon and Breach, New York, 1970) 59-61. MR 41 #8260; Z 263.05018. J.W. DiPaola. Configurations in small hyperbolic planes. Ann. N.Y. Acad. Sci. 175 (1970) 93-103. MR 42 #6704: Z 244.50010. J.W. DiPaola. Some finite point geometries, Math. Mag. 50 (1977) 79-83. Z 376.05013. J.W. DiPaola and E. Nemeth. Generalized triple systems and medial quasigroups, Proc. 7th South-Eastern Conf. Combinatorics. Graph Theory and Computing (Baton Rouge, 1976), Congressus Numerantium XVlI (Utilitas Math. Winnipeg, 1976) 289-306. MR 55 #2599: Z 359.20072. LD28.11 J.W. DiPaola and E. Nemeth, Applications of parallelisms, Ann. N . Y. Acad. Sci. 319 (1979) 153-163. ID291 A.C. Dixon. Note o n Kirkman's problem. Messenger Math. 23 (1893-94) 88-89.
Bibliography and survey of Sreiner sysrems
325
D.H. Doehlert, Balanced sets of balanced incomplete block designs of block size three, Technometrics 7 (1965) 561-577. A.P. Domorjad, MatematiEeskije igry i razvleEenija (Fizmatgiz, Moskva, 1961) Chapter 20: Sostavlenije raspisanij, pp. 109-112. MR 23A #761. (English translation: Mathematical Games and Pastimes (Pergamon, Oxford, 1964) Z 116, p. 1. J.R. Doner, CIP neofields and combinatorial designs, Notices Amer. Math. SOC.19 (1972) A-388 J. Doyen, Sur la structure de certains systtmes triples de Steiner, Math. Z. 111 (1969) 289-300. MR 40 #53; Z 182, p. 27. J. Doyen, On the number of non-isomorphic Steiner systems S(2, m, n). Combinat. Structures and their Applications (Proc. Confer. Calgary 1969) (Gordon and Breach, New York, 1970) 63-64. Z 247.05010. J. Doyen, Problem 30, Combinat. Structures and their Applications (Proc. Confer. Calgary 1969) (Gordon and Breach, New York, 1970) 504. J. Doyen, Sur la croissance du nombre de systbmes triples de Steiner non isomorphes, J. Combinatorial Theory 8 (1970) 424-441. MR 41 #1555; Z 192. p. 333. J. Doyen, Systtmes triples de Steiner non engendrts par tous leurs triangles, Math. Z. 118 (1970) 197-206. MR 43 #70; Z 201, p. 533. J. Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1971-72) 315-319. MR 45 #8542; Z 272.05013. J. Doyen, Constructions of disjoint Steiner triple systems, Proc. Amer. Math. Soc. 32 (1972) 409-416. MR 45 #4990; Z 215, p. 335. J. Doyen, Recent results on Steiner triple systems, Finite Geometric Structures and their Applications (C.I.M.E., I1 Ciclo Bressanone 1972) (Edizioni Cremonese, Roma, 1973) 201-210. MR 49 #10567; Z 264.05008. J, Doyen, Recent developments in the theory of Steiner systems, Teorie Combinatorie, I, Colloq. Internaz., Atti dei Convegni Lincei 17 Accademia Nazionale dei Lincei, Roma, 1976) 227-285. MR 55 #10286; Z 348.05001. J. Doyen and X. Hubaut, Finite regular locally projective spaces, Math. Z. 119 (1971) 83-88. MR 43 #3897; Z 199, p. 241; Z 202, p. 194. J. Doyen, X. Hubaut and M. Vandensavel, Ranks of incidence matrices of Steiner triple systems, Math Z. 163 (1978) 251-259. Z 373.05011; Z 383.05007. J. Doyen and A. Rosa, A bibliography and survey of Steiner systems, Bollet. U.M.I. (4) 7 (1973) 392-419. MR 48 #118; Z 273.05001. [D44.1] J. Doyen and A. Rosa, An extended bibliography and survey of Steiner systems, Proc. 7th Manitoba Conf. Numerical Math. and Computing, Congressus Numerantium XX (Utilitas Math., Winnipeg, 1978) 297-361. [D45] J. Doyen and G. Valette, On the number of nonisomorphic Steiner triple systems, Math. Z. 120 (1971) 178-192. MR 43 #4695; Z 203, p. 308; Z 208, p. 24. [D46] J. Doyen and M. Vandensavel, Non isomorphic Steiner quadruple systems, Bull. SOC.Math. Belg. 23 (1971) 393-410. MR #117; Z 252.05012. [D47] J. Doyen and R.M. Wilson, Embeddings of Steiner triple systems, Discrete Math. 5 (1973) 229-239. MR 48 #588l; Z 263.05017. [D48] L.M.H.E. Driessen, G.H.M. Frederix and J.H. van Lint, Linear codes supported by Steiner triple systems, Ars Combinatoria 1 (1976) 33-42. MR 54 #7106; Z 337.94006. lD48.11 P. Ducrocq and F. Sterboul, On G-triple systems, Publications du Laboratoire de Calcul de I’Universitt des Sciences et Techniques de Lille, No. 103, Janvier 1978, 18 pp. [D49] H.E.Dudeney, On Kirkman’s schoolgirl problem, Educational Times Reprints (2) 14 (1908) 97-99; 15 (1909) 17-19; 17 (1910) 35-38, 53. [D50] P. Duhem, Notice sur la vie et les travaux de Georges Brunel (1856-1900). Mtm. SOC.Sci. Phys. Nat. Bordeaux (6) 2 (1902) I-XCI. 0. Eckenstein, Note on Kirkman’s schoolgirl problem, Educational Times Reprints (2) 16 [El] (1909) 76-77; 17 (1910) 38-39, 49-53.
326
J. Doyen, A. Rosa
0.Eckenstein, Bibliography of Kirkman’s schoolgirl problem, Messenger Math. (2) 41
rF4.11
[F7]
lF91
(1911-12) 33-36. J. Edmonds, see [Y4]. M. Eggen, see [C13]. R.C. Ehrmann, Kirkmans schoolgirls as points in a finite geometry, Notices Amer. Math. SOC.20 (1973) A-530. R.C. Ehrmann, Projective space walk for Kirkman’s school-girls (to appear). K. von Eichhorn, Steiner Systeme, Codes und affine Raume, Diplomarbeit, Arbeitsberichte des Instituts fur Mathematische Maschinen und Datenverarbeitung, Bd. 10, Nr. 11, Erlangen. Juli 1977. 94 pp. A. Emch, Triple and multiple systems, their geometric configurations and groups, Trans. Amer. Math. SOC.31 (1929) 25-42. A. Emch, Finite groups and their geometric representation, J. reine angew. Math. 162 (1930) 238-2.55. A. Emch. Ueber einer besondere Klasse von algebraischen Flachen, Comment. Math. Helv. 2 (1930) 99-1 15. P. Erdos and H. Hanani. On a limit theorem in combinatorial analysis, Publ. Math. Debrecen 10 (1963) 10-13. MR 29 #3394; Z 122, p. 148. P. Erdos and A. Renyi. On some combinatorial problems, Publ. Math. Debrecen 4 (195556) 398-405. MR 18. p. 3; Z 70, p. 11. I.M.H. Etherington, Quasigroups and cubic curves, Proc. Edinburgh Math. SOC. (2) 14 1196465) 273-291. MR 33 #4170; Z 133, p. 275. T. Evans. see [L30]. T. Evans, Universal-algebraic aspects of combinatorics (to appear). G. Ferrero. Gruppi di Steiner e sistemi fini, Matematiche 27 (1972) 167-190. MR 48 #1944; Z 264.0501 1. G. Ferrero. Sui gruppi che amettono funzioni di Steiner, Rend. 1st. Mat. Univ. Trieste 4 (1972) 156-170. MR 48 #5883; Z 262.20045. G. Ferrero, Deformazioni, raffinamenti e composizioni di funzioni di Steiner (I), Riv. Mat. 1Jniv. Parma (3) 1 (1972) 125-142. MR 51 #2937; Z 344.05010. G . Ferrero. Deformazioni, raffinamenti e composizioni di Steiner ( I t ) , Riv. Mat. Univ. Parma (4) 3 (1977) 151-157. G. Ferrero, Su un problema relativo ai sistemi di Steiner disgiunti. Rend. 1st. Mat. Univ. Trieste 7 (1975) 1-7. MR 52 #13424; Z 319.05009. G. Ferrero and G . Gallina, Funzioni di Steiner con parecchi moltiplicatori, Riv. Mat. Univ. Parma (4) 4 (1978) 475-484. G. Ferrero and A. Suppa, Sistemi, annclloidi e funzioni di Steiner, Atti Sem. Mat. Fiz. Univ. Modena 20 (1971) 272-280. Z 249.05001. B. Fischer. Eine Kennzeichnung dcr symmetrischen Gruppen vom Grade 6 und 7. Math. Z. 95 (1967) 288-298. MR 34 #5909; Z 139. p. 251. R.A. Fisher. An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugenics 10 (1940) 52-75. MR 1. p. 348. K.A. Fisher, New cyclic solutions to problems in incomplete blocks, Ann. Eugenics 11 (1942) 290-299. MR 4,p. 21. R.A. Fisher and F. Yates, Statistical tables for biological, agricultural and medical research, Oliver and Boyd, Edinburgh 1938, 5th ed. 1957. 25-29, 90-93. Z 79, p. 352. (6th ed. Hafner, New York, 1963). T.S. Fiske. Frank Nelson Cole, Bull. Amer. Math. SOC.33 (1927) 773-777. F. Fitting, Die Herstellung von Tripelsystemen fur jede Anzahl von Elementen, Nieuw Arch. Wisk. (2) 9 (1911) 359-369. F. Fitting. Reitrage zum Steincr‘schen Problem, Nieuw Arch. Wisk. (2) 10 (1913) 88-99. F, Fitting, Aufstellung aller voneinander unabhangigen Losungen des Kirkman’schen 15Pensionatsdamen-Problems, Nieuw Arch. Wisk. (2) 10 (1913) 244-251. F. Fitting, Zyklische Liiwngen des Steiner’schen Problems, Nieuw Arch. Wisk. (2) 1 1 (1915) 140- 148.
Bibliography and suruey of Sfeiner systems
[G0.1] [G0.2] [G0.3] [GI1 [G21 [G31 [G3.1]
[G3.2]
rG6.11
327
M.K. Fort and G.A. Hedlund, Minimal coverings of pairs by triples, Pacific J. Math. 8 (1958) 709-719. MR 21 #2595; Z 84, p. 14. W.J. Frascella, A generalization of Sierpinski’s theorem on Steiner triples and the axiom of choice, Notre Dame J. Formal Logic 6 (1965) 163-179. MR 33 #2552. W.J. Frascella, The construction of a Steiner triple system on sets of the power of the continuum without the axiom of choice, Notre Dame J. Formal Logic 7 (1966) 196-202. MR 35 #2751; Z 147, p. 262. W.J. Frascella, Combinatorial designs on infinite sets, Notre Dame J. Formal Logic 8 (1967) 27-47. MR 38 #4339; Z 189. p. 296. W.J. Frascella, Non-existence of a certain combinatorial design on an infinite set, Notre Dame J. Formal Logic 10 (1969) 317-323. MR 40 #1284; Z 206, p. 20. G.H.M. Frederix, see [D48]. A.H. Frost, General solution and extension of the problem of the fifteen schoolgirls, Quart. J. Pure Appl. Math. 11 (1871) 26-37. D.R. Fulkerson, G.L. Nemhauser and L.E. Trotter Jr., Two computationally difficult set covering problems that arise in computing the 1-width of incidence matrices of Steiner triple systems, Math. Program. Study 2 (North-Holland, Amsterdam, 1974) 72-81. Z 353.90060. G. Gallina, Su certi sistemi di Steiner non disgiunti, Riv. Mat. Univ. Parma (4) 3 (1977) 199-202. G . Gallina, Gruppi ammettenti funzioni di Steiner, Bollet. U.M.I. (5) 15 (1978) 225-230. Z 387.05001. G. Gallina, Sull’esistenza di certc funzioni di Steiner, Riv. Mat. Univ. Parma (4) 4 (1978) 2 15-21 9. B. Canter, Endliche Vervollstandigung endlicher partieller Steinerscher Systeme, Arch. Math. 22 (1971) 328-332. MR 45 #3218; Z 221.05050. B. Canter, A catalogue of Steiner systems, Technische Hochschule Darmstadt Preprint Nr. 7, March 1972. B. Canter. Finite partial quadruple systems can be finitely embedded, Discrete Math. 10 (1974) 397-400. MR 50 #1912; Z 291.05018. B. Canter, A. Mathon and A. Rosa, A complete census of (10. 3,2)-block designs and of Mendelsohn triple systems of order ten. I. Mendelsohn triple systems without repeated blocks, Proc. 7th Manitoba Conf. Numerical Math. and Computing, Winnipeg 1977, Congressus Numerantium XX (Utilitas Math., Winnipeg, 1978) 383-398. B. Canter and R. Metz, Kombinatorische Algebra: Koordinatisierung von Blockplanen, Beitrage zur geometrischen Algebra (Proc. Sympos. Duisburg 1976), Lehrbucher u. Monographien Exakt. Wiss. Math. Reihe, Bd. 22 (Birkhauser, Basel, 1977) 111-124. B. Canter, J. Pelikan and L. Teirlinck, Small sprawling systems of equicardinal sets, Ars Combinatoria 4 (1977) 133-142. MR 57 #9553. B. Canter and H. Werner, Equational classes of Steiner systems, Algebra Univ. 5 (1975) 125-140. MR 53 #7908a; Z 312.08002. B. Canter and H. Werner. Equational classes of Steiner systems II., Proc. Conf. Algebraic Aspects of Combinatorics (Toronto, 1975), Congressus Numerantium XI11 (Utilitas Math. Winnipeg, 1975) 283-285. MR 53 #7908b; Z 327.08007. M. Gardner, On the remarkable Csaszir polyhedron and its applications in problem solving, Scientific American 232 (5) (1975) 102-107. A.M.H. Gerards, see [B25.1]. P.B. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. Thesis, University of Toronto 1976. (=Department of Computer Science, University of Toronto, Technical Report No. 92, May 1976). P.B. Gibbons and R. Mathon, On a new class of Steiner quadruple systems on 16 symbols, Proc. Conf. Algebraic Aspects of Combinatorics (Toronto, 1975) Congressus Numerantium XI11 (Utilitas Math., Winnipeg, 1975) 227-232. MR 52 #7922; Z 334.05020. P.B. Gibbons, R. Mathon and D.G. Corneil, Steiner quadruple systems on 16 symbols, Proc. 6th South-Eastern Conf. Combinatorics, Graph Theory and Computing (Boca Raton, 1975). Congressus Numerantium XIV (Utilitas Math., Winnipeg, 1975) 345-365. MR 53 #5327; Z 326.05012.
J. Doyen, A. Rosa P.B. Gibbons. R. Mathon and D.G. Corneil, Computing techniques for the construction and analysis of block designs, Utilitas Math. 11 (1977) 161-192. Z 352.05009. J.M. Goethals, On the Golay perfect binary code, J. Combinatorial Theory (A) I 1 (1971) 178-186. MR 46 #8755. J.M. Goethals and J.J. Seidel, Quasisymmetric block designs, Combinat. Structures and their Applications (Proc. Confer. Calgary 1969) (Gordon and Breach. New York, 1970) 111-1 16. MR 42 #5818; Z 251.05009. J.M. Goethals and J.J. Seidel, Strongly regular graphs derived from combinatorial designs, Canad. J. Math. 22 (1970) 597-614. MR 44 #106; Z 198, p. 293. M.G. Goudarzi. Sur le probleme de Kirkman, Acad. Roy. Belg. Bull. CI. Sci. 53 (1967) 183-199. MR 35 #6019; Z 146, p. 417. M.G. Goudarzi, Sur le probltme de Kirkman gtniralisk. Acad. Roy. Belg. Bull. C1. Sci. 53 (1967) 1379-1384. MR 38 #6444; Z 155, p. 299. [G15.1] M.J. Grannel and T.J. Griggs, A note on the Steiner systems S(S,6,24), Ars Combinatoria (to appear). [G16] L.M. Graves, A finite Bolyai-Lobachevsky plane, Amer. Math. Monthly 69 (1962) 130-132. MR 26 #4252; Z 106, p. 143. [G16.1] D.L. Greenwell and C.C. Lindner, Some remarks on resolvable Steiner quadruple systems, Ars Combinatoria 6 (1978) 215-221. T.J. Griggs. see [G15.l]. E.J. Grinberg and B.A. Ikauniece, 0 postrojenii polnych sistem trojek Stejnera, Voprosy Kibernetiki 16 (1975). Moskva (1975) 21-29, B.H. Gross. Intersection triangles and block intersection numbers of Steiner systems. Math. Z. 139 (1974) 87-104. MR 50 #9614; Z 276.05018; Z 285.05021. K.B. Gross, On the maximal number of orthogonal Steiner triple systems, Proc. Conf. Algebraic Aspects of Combinatorics (Toronto, 1975). Congressus Numerantium XI11 (Utilitas Math., Winnipeg, 1975) 233-240. MR 52 #7923; Z 326.0501 1. K.B. Gross, On the maximal number of pairwise orthogonal Steiner triple systems, J. Combinatorial Theory (A) 19 (1975) 256-263. MR 52 #2917; Z 334.05016. R. Guerin, Vue d’ensemble sur les plans en blocs incomplets Cquilibrts et partiellement tquilibres. Rev. Inst. Internat. Statist. 33 (1965) 24-58. MR 31 #4732; Z 137, p. 374. M. Guregova and A. Rosa, Using the computer to investigate cyclic Steiner quadruple systems. Mat. casopis 18 (1968) 229-239. MR 39 #80; Z 169, p. 20. M. Guza, see [A24]. R.K. Guy, A many-facetted problem of Zarankiewicz, The Many Facets of Graph Theory (Springer. Berlin, 1969) 129-1423, MR 41 #91: Z 186, p. 18. R.K. Guy, Twenty odd questions in combinatorics, Combinatorial Mathematics and its Applications (Proc. 2nd Chapel Hill Conf.), Univ. of N.C. (1970) 209-237. MR 42 #1666; Z 209. p. 210. R.K. Guy, see [B34]. G. Haggard. On the function N(3,2. u ) , Proc. 3rd South-Eastern Conf. Combinatorics, Graph Theory and Computing (Boca Raton, 1972) 243-250. MR 50 #121; Z 265.05005. J.I. Hall, Steiner triple systems with geometric minimally generated subsystems, Quart. J. Math. Oxford Ser. (2) 25 (1974) 41-50. MR 49 #10569; Z 279.05017. J.I. Hall and J.T. Udding, On pairs of Steiner triple systems intersecting in subsystems, Technological University Eindhoven Report 76-WSK-04, August 1976, 22 pp. Z 3Sl.O5010. J.1. Hall and J.T. Udding. On intersection of pairs of Steiner triple systems, Indag. Math. 39 1177) ( = Proc. Koninkl. Nederl. Akad. Wetensch. Ser. A 80 (1977)) 87-100. MR 56 #5317: Z 354.05027. M. Hall Jr., Uniqueness of t h e projective plane with 57 points, Proc. Amer. Math. SOC.4 (1053) 912-916. MR 15 p. 460; 2 52, p. 165. Correction: Proc. Amer. Math. SOC.S (1954) 904-997. 2 S6, p. 385. M. Hall Jr., A survey of combinatorial analysis, Some Aspects of Analysis and Probability (Wiley. New York, 1958) 35-104. MR 22 #2556; Z 86. p. 12. (Russian translation: Kombinatornyj analu (Izdat. inostr. lit., Moskva, 1963) Z 192. p. 89.)
Bibliography and suruey of Sfeiner systems
329
M. Hall Jr., Automorphisms of Steiner triple systems, IBM J. Res. Develop. 4 (1960) 460-472. MR 23A #1282; Z 100, p. 18. M. Hall Jr., Current studies on combinatorial designs, Proc. Symp. Appl. Math. 10 (Amer. Math. Soc. Providence, RI, 1960) 1-14. MR 25 #4020; 2 96, p. 122. M. Hall Jr., Automorphisms of Steiner triple systems, Proc. Symp. Pure Math. 6 (Amer. Math. SOC.Providence, RI, 1962) 47-66. MR 24A #2890; 2 114, p. 12. M. Hall Jr., Discrete variable problems, Survey of Numerical Analysis (McGraw-Hill, New York, 1962) 518-542. MR 24A #1837. M. Hall Jr., Block designs, in; E.F. Beckenbach, ed., Applied Combinatorial Mathematics (Wiley, New York, 1964) 369-405. Z 158, p. 16. M. Hall Jr., Numerical analysis of finite geometries, Proc. IBM Sci. Comput. Sympos. Combinatorial Problems, IBM Data Processing Division, White Plains, NY (1966) 11-22. MR 36 #5812. M. Hall Jr.. Group theory and block designs, Proc. Internat. Conf. Theory of Groups (Canberra, 1965) (Gordon and Breach, New York, 1967) 115-144. MR 36 #2514; Z 323.2011. (Russian translation: Teorija grupp i blok-schemy, Kombinatornyj analiz 1 (1971) 10-34. Z 285.20006.) M. Hall Jr., Combinatorial Theory, Blaisdell, Waltham, Mass. 1967. MR 37 #80; Z 196, p. 24. (Russian translation: Kombinatorika, Mir, Moskva 1970. MR 42 #4411.) M. Hall Jr., Construction of combinatorial designs, Teorie Combinatorie, 1, Colloq. Intcrnaz., Atti dei Convegni Lincei 17 (Accademia Nazionale dei Lincei, Roma, 1976) 13-41. MR 55 #12529. M. Hall Jr. and D.E. Knuth, Combinatorial analysis and computers, Amer. Math. Monthly 72 (1965) 11, 21-28. MR 30 #3030; Z 127, p. 90. M. Hall Jr. and J.D. Swift, Determination of Steiner triple systems of order 15, Math. Tables Aids Comput. 9 (1955) 146-152. MR 18, p. 192. M. Hall Jr., J.D. Swift and R. Killgrove, On projective planes of order nine, Math. Tables Aids Comput. 13 (1959) 233-246. MR 21 #5933; Z 94, p. 337. M. Hall Jr., J.D. Swift and R. Walker, Uniqueness of the projective plane of order eight, Math. Tables Aids Comput. 10 (1956) 186-194. MR 18, p. 816; 2 73, p. 365. H. Hanani, On quadruple systems, Canad. J. Math. 12 (1960) 145-157. MR 22 #2558; Z 92, p. 12. H. Hanani, A note on Steiner triple systems, Math. Scand. (1960) 154-156. MR 23A #2330; Z 100, p. 18. H. Hanani, The existence and construction of balanced incomplete block designs, Ann. Math. Statist. 32 (1961) 361-386. MR 29 #4161; Z 107, p. 361. H. Hanani, On some tactical configurations, Canad. J. Math. 15 (1963) 702-722. MR 28 #1136; Z 196, p. 291. H. Hanani, On covering of balanced incomplete block designs, Canad. J. Math. 16 (1964) 615-625. MR 29 #4706; 2 192, p. 92. H. Hanani, A balanced incomplete block design, Ann. Math. Statist. 36 (1965) 711. MR 30 #4358; 2 131, p. 182. H. Hanani, On balanced incomplete block designs and related designs, Technion’s Preprint Series No. MR-7, Technion, Israel Institute of Technology, Haifa, 1968. H. Hanani, On balanced incomplete block designs with blocks having five elements, J. Combinatorial Theory (A) 12 (1972) 184-201. MR 45 #6645; Z 247.05009. H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11 (1975) 255-369. MR 52 #2918; Z 361.62067. H. Hanani, A class of three-designs, Technion Preprint Series No. MT-324, November 1976, 21 PP. H. Hanani, D.K. Ray-Chaudhuri and R.M. Wilson, On resolvable designs, Discrete Math. 3 (1972) 343-357. MR #8857; 2 263.05016. H. Hanani, see rE91. . . [H30.1] A. Hartman, Resolvable Designs, MSc. Thesis, Technion, Haifa, June 1978. rH30.21 A. Hartman, Parallelism of Steiner quadruple systems, Ars Combinatoria 6 (1978) 27-37.
1.Doyen, A. Rosa H. Hasse. Gruppentheoretischer Beweis eines Satzes uber gewisse Tripelsysteme, Norsk Mat. Tidsskr. 13 (1931) 195-107. Z 3, p. 385. S.H. Heath, The existence of finite Bolyai-Lobachevsky planes, Math. Mag. 43 (1970) 244-249. MR 45 #7584; Z 225.50012. S. Heath and G.R. Wylie, Some observations on BL(3,3), Univ. Nac. Tucuman Rev. Ser. A 20 (1970) 117-123. MR 44 #6524. A. Hedayat, A set of three orthogonal Latin squares of order 15 associated with a Kirkman-Steiner triple system of orderl5, Ann. Math. Statist. 42 (1971) 1803. A. Hedayat, An algebraic property of the totally symmetric loops associated with KirkmanSteiner triple systems, Pacific J. Math. 40 (1972) 305-310. MR 46 #3340; 2 241.05014. A. Hedayat and B.L. Raktoe, On the equivalence of resolvable Kirkman-Steiner triple systems and sets of mutually orthogonal Latin squares, Ann. Math. Statist. 41 (1970) 1140. G.A. Hedlund, see [FlS]. L. Heffter, Ueber Nachbarconfigurationen, Tripelsysteme und metacyklische Gruppen, Deutsche Mathem. Vereinig. Jahresber. 5 (1896) 67-68. L. Heffter, Ueber Tripelsysteme, Math. Ann. 49 (1897) 101-112. W. Heise, Bericht iiber I-affine Geometrien, J. Geometry 1 (1971) 197-224. MR 47 #5702: Z 228.50033. W. Heise. Es gibt keine 4-afflne Ebene der Ordnung 4, J. reinc angew. Math. 252 (1972) 104-106. MR 45 #7585; Z 226.05013. W. Heise and T. Matzner, List of admissible parameters for Steiner systems, TU Munchen, lnstitut fur Mathematik. TUM-MATH-7724, Juni 1977, 122 pp. W. Heise and J. Timm, I-affine Raume, Manuscripta Math. 4 (1971) 31-37. MR 45 #4251; Z 203, p. 530. M. Henderson, Generalized finite planes, Colloq. Math. 13 (1964) 29-36. MR 30 #5210; Z 125, p. 100. M. Henderson. Finite Bolyai-Lobachevsky k-spaces, Colloq. Math. 15 (1966) 205-210. MR 34 #1909: Z 146, p. 161. M.T. Hermoso, see [A14]. D.G. Higman and C.C. Sims, A simple group of order 44,352,000, Math. Z. 105 (1968) 110-1 13. MR 37 #2854; Z 186, p. 40. A.J.W. Hilton, On Steiner and similar triple systems, Math. Scand. 24 (1969) 208-216. MR 41 #3296: Z 188, p. 40. A.J.W Hilton, A simplification of Moore’s proof of the existence of Steiner triple systems, J. Combinatorial Theory (A) 13 (1972) 422-425. MR 46 #1616; Z 244.05010. A.J.W. Hilton, On the Szamkolowicz-Doyen classification of Steiner triple systems, Proc. London Math. Soc. (3) 24 (1977) 102-116. MR 55 #128. [H48.1] A.J.W. Hilton, Dimension in linear spaces, Problkmes Combinatoires et Thkorie des Graphes, Colloq. Internat. C.N.R.S. No. 260 (Editions du CNRS, Paris. 1978) 233-236. A.J.W. Hilton, see [A12.2] W491 J.W.P. Hirschfcld. Cyclic projectivities in PG(n. 4).Teorie Combinatorie, 1. Colloq. Internaz.. Atti dei Convrgni Lincci 17 (Accademia Nazionale dei Lincei, Roma, 1976) 201-211. Z 359.50022. [HSO] A.J. Hoffman and M. Richardson, Block design games, Canad. J . Math. 13 (1961) 110-128. MR 23B #2067; Z 101. p. 372. [ H s l ] J. Homer, On triads of once-paired elements, Quart. J. Pure Appl. Math. 9 (1868) 15-18, [H52] J.D. Horton, Variations on a theme by Moore, Proc. Louisiana Conf. Combinatorics, Graph Theory and Computing, lmuisiana State Univ.. Baton Rouge (1970) 146-166. MR 46 #8864; 2 248.05018. [H53] J.D. Horton, R.C. Mullin and Stanton R.G.. Minimal coverings of pairs by quadruples, Proc. 2nd Louisiana Conf. Combinatorics. Graph Theory and Computing (Baton Rouge. 197 I ) 495-516. MR 47 #6518; Z 289.05023. (H5J] N.C. Hsu, Note on certain combinatorial designs, Proc, Amer. Math. SOC. 13 (1962) 682-685. MR 25 #5004; Z 111. p. 155. [ H 5 5 ] X. Hubaut. Limitation du nombre de points d’un (k,n)-arc rkgulier d’un plan projectif fini,
Bibliography and suruey of Steiner systems
w41 [K4.1]
331
Atti Acad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur. (8) 48 (1970) 490-493. MR 44 #5856; Z 204, p. 211. X. Hubaut, Caractirisation de certains systbmes de Steiner, Atti del Convegno di Geometria Combinatoria e sue Applicazioni (Perugia 1970). Universita degli Studi di Perugia (1971) 295-296. X. Hubaut, Systbmes de Steiner minimaux, Bull. SOC.Math. Belg. 23 (1971) 411-415. MR 48 #120; Z 282.05013. X. Hubaut, see [D42], [D43]. D.R. Hughes, Combinatorial analysis, t-designs and permutation groups, Proc. Symp. Pure Math. 6 (Amer. Math. Soc., Providence, RI, 1962) 39-41. MR 24A #3206; Z 114, p. 12. D.R. Hughes, On 1-designs and groups, Amer. J. Math. 87 (1965) 761-778. MR 32 #5727; Z 134, p. 30. D.R. Hughes, see [D7]. S.H.Y. Hung and N.S. Mendelsohn, Directed triple systems, J. Combinatorial Theory (A) 14 (1973) 310-318. MR 47 #3190; Z 263.05020. S.H.Y. Hung, see [M20]. J.F. Hurley and A. Rudvalis, Finite simple groups, Amer. Math. Monthly 84 (1977) 693-714. F.K. Hwang and S. Lin, A direct method to construct triple systems, J. Combinatorial Theory (A) 17 (1974) 84-94. MR 52 #127; Z 282.05010. B.A. Ikauniece, see [G17]. R.W. Irving, Generalized Ramsey numbers for small graphs, Discrete Math. 9 (1974) 251-264. MR 51 #12591; Z 287.05104. L.I. Istomina, Krugovaja ploskost' porjadka 5, Kombinatornyj Analiz 2 (1972), Izdat. Mosk. Univ., Moskva (1972) 73-85. R.K. Jain, On cyclic Steiner quadruple systems, M.Sc. Thesis, McMaster University, Hamilton, 1971. R.K. Jain, On nonisomorphic Steiner quadruple systems, Notices Amer. Math. SOC. 20 (1973) A-514. E.C. Johnsen and T. Storer, Combinatorial structures in loops, IV. Steiner triple systems in property cyclic neofields of prime-power order, Pacific J. Math. 52 (1974) 115-127. MR 53 #12982; Z 256.05008; Z 286.05007. E.C. Johnsen and T. Storer, Combinatorial structures in loops, IV. Steiner triple systems in neofields, Math. Z. 138 (1974) 1-14. MR 56 #5319; Z 276.05019; Z 285.05022. D.M. Johnson and N.S. Mendelsohn, Extended triple systems, Aequat. Math. 8 (1972) 291-298. MR 47 #3211; Z 256.20096. W. Jonsson, On the Mathieu groups M,,,M,,, M,, and the uniqueness of the associated Steiner systems, Math. Z. 125 (1972) 193-214. MR 45 #4264; Z 221.20015. W. Jonsson, The (56, 11.2) design of Hall, Lane and Wales, J. Combinatorial Theory (A) 14 (1973) 113-118. MR 46 #7050; Z 257.05016. W. Jonsson and J. McKay, More about the Mathieu group M,,. Canad. J. Math. 28 (1976) 929-937. MR 55 #139; Z 319.20005. J.G. Kalbfleisch and R.G. Stanton, Maximal and minimal coverings of (k - 1)-tuples by k-tuples, Pacific J. Math. 28 (1968) 131-140. MR 38 #2032; Z 182, p. 28. J.G. Kalbfleisch, see [S40], [S41], [S42]. W.M. Kantor, On 2-transitive groups in which the stabilizer of two points fixes additional points, J. London Math. Soc. 5 (1972) 114-122. MR 45 #6907. P.B. Kaufmann, Studien iiber zyklische Dreiersysteme der Form N = 6n + 3, InauguralDissertation der Math. -Natur. Fakultat der Universitat Freiburg in der Schweiz, Sarnen 1926. A.D. Keedwell, see [D8]. O.H. Kegel and A. Schleiermacher, Amalgams and embeddings of projective planes, Geometriae Dedicata 2 (1973) 379-395. MR 49 #3678; Z 271.50017. T. Kepka, Distributive Steiner quasigroups of order 3'. CMUC 19 (1978) 389-401. Z 373.20062. R. Killgrove, see [Hl8].
J . Doyen, A. Rosa
J.D. King, A characterization of some doubly transitive groups, Math. 2. 107 (1968) 43-48. Correction: 112 (1969) 3 9 S 3 9 4 . MR 41 #312; 2 164, p. 25. T.P. Kirkman, On a problem in combinations, Cambridge and Dublin Math. J. 2 (1847) 191-204. T.P. Kirkman. Query VI.. Lady’s and Gentleman’s Diary (1850) 48. T.P. Kirkman, Note on an unanswered prize question, Cambridge and Dublin Math. J. 5 (1850) 255-262. T.P. Kirkman, On the triads made with fifteen things. London, Edinburgh and Dublin Philos. Mag. and J. Sci. (3) 37 (1850) 169-171. T.P. Kirkman. Solutions to Query VI., Lady’s and Gentleman’s Diary (1851) 48. T.P. Kirkman. Theorems on combinations, Cambridge and Dublin Math. J. 8 (1853) 38-45. T.P. Kirkman. On the perfect r-partitions of N = r 2 - r + 1, Transactions of the Historic Society of Lancashire and Cheshire 9 ( 1 8 5 6 5 7 ) 127-142. T.P. Kirkman, On the puzzle of the fifteen young ladies, London, Edinburgh and Dublin Philos. Mag. and J. Sci. (4) 23 (1862) 198-204. T.P. Kirkman. Solutions of three problems proposed by W. Lea, Educational Times Reprints I 1 (1869) 97-99. T.P. Kirkman. Solution of two questions (3143 and 4005), Educational Times Reprints 19 (1871) 71-72; 20 (1874) 36. T.P. Kirkman and H.J. Woodall, Solution of a problem, Educational Times Reprints 55 ( 189 1) 60. M.S. Klamkin and D.J. Newman, Some combinatorial problems of arithmetic, Math. Mag. 42 (1969) 52-56. MR 37 #78; Z 186. p. 79. [K17.1] S. Klossek, Kommutative Spiegelungsraume, Mitt. Math. Sem. Giessen, Heft 117. Giessen 1975. 72 pp. MR 53 #10966; 2 317.20043. D.E. Knuth. see [H16]. [KIK] E. Kohler, Eine kombinatorische Aufgabe. Mitt. Math. Ges. Hamburg 10 (1973) 130-131. MR 52 #2919. [K19] E. Kohler. Bemerkungen iiber Steiner-Systeme, Abh. Math. Sem. Univ. Hamburg 40 (1974) 226-228. MR 48 #8261; Z 277.05137. [K20] E. KBhler, Zur Theorie der Steiner-Systeme, Abh. Math. Sem. Univ. Hamburg 43 (1975) 181-185. MR 51 #10122; Z 321.05015, [K21] E. Kohler. Numerische Existenzkriterien in der Kombinatorik. Numerische Methoden bei graphen-theoretischen und kombinatorischen Problemen (Birkhauser, Easel. 1975) 99-108. MR 54 #5005; Z 326.05014. [ K 2 ] E. Kohler, Unendliche gefaserte Steinersysteme. J. Geometry 9 (1977) 73-77. MR 56 #5320. 1K22.11 E. Kohler, Zyklische Quadrupelsysteme, Abh. Math. Sem. Univ. Hamburg 48 (1979) 1-24. . . [K23] E. Kiihler and W.-L. Piotrowski, Einige Bemerkungen zur Theorie der f-Designs, Methods of Operations Research XXIII, V111. Obenvolfach-Tagung iiber Operations Research, August 1976 (Verlag Anton Hain, Meisenheim am Clan, 1977) 108-122. lK241 .I. Konvalina. Cyclic projective planes, planar difference sets and non-associative algebras, Ph.D. Thesis. SUNY at Buffalo 1975, 61 pp. [KZS] A. Kotzig, C.C. Lindner and A. Rosa. Latin squares with no subsquares of order two and disjoint Steiner triple systems, Utilitas Math. 7 (1975) 287-294. MR 53 #5331: 2 31 1.05013. [K%] A. Kotzig and A. Rosa, Nearly Kirkman systems, Proc. 5th South-Eastern Conf. Cornbinatorics. Graph Theory and Computing (Boca Raton, 1974). Congressus Numerantiurn X (Utilitas Math. Winnipeg, 1974) 607-614. MR 51 #190; Z 311.05023. (K271 G. Kowalewski. Alte und neue mathematische Spiele. Eine Einfiihrung in die Unterhaltungsmathematik (B.G. Teubner, Leipzig. 1930) Kap 1V. 13: Das Problem der fiinfzehn Pensionatsdamen. [K2X] M. Kra’itchik, La mathtmatique des jeux ou rtcriations mathematiques, Stevens, Bruxelles 1930. Chap. 18: Jeux de ripartition, pp. 540-555. [ K29] M. Kraytchik, Mathematical Recreations (Norton, New York, 1942) (Reprinted by Dover, New York 1953. MR 14, p. 620.)
Bibliography and survey of Steiner systems
333
E.S. Kramer, Indecomposable triple systems, Discrete Math. 8 (1974) 173-180. MR 48 #10863; Z 276.05020. E.S. Kramer, Some triple system partitions for prime powers, Utilitas Math. 12 (1977) 113-116. Z 367.05010. ESKramer, A generalized Room square GRS(4.24) of dimension 9, Discrete Math. 20 (1977) 91-92. MR 57 #9516; Z 377.05011. E.S. Kramer and S.S. Magliveras, Some mutually disjoint Steiner systems, J. Combinatorial Theory (A) 17 (1974) 39-43. MR 49 #7155; Z 282.05012. E.S. Kramer and D.M. Mesner, Intersections among Steiner systems, J. Combinatorial Theory (A) 16 (1974) 273285. MR 49 #78; Z 282.05011. E.S. Kramer and D.M. Mesner, On Steiner tableaus, Discrete Math. 10 (1974) 123-131. MR 50 #1933; Z 305.05014. E.S. Kramer and D.M. Mesner, The possible (impossible) systems of 11 disjoint S(2,3, 13)’s (S(3,4,14)’s) with automorphism of order 11, Utilitas Math. 7 (1975) 55-58. MR 51 #7898; Z 309.05014. E.S. Kramer and D.M. Mesner, Admissible parameters for Steiner systems S(r, k, u ) with a table for all (o-r)<498, Utilitas Math. 7 (1975) 211-222. MR 51 #7899; Z 309.05012. J.F. Lawless, Painvise balanced designs and the construction of certain combinatorial systems, Proc. 2nd Louisiana Conf. Combinatorics, Graph Theory and Computing (Baton Rouge, 1971) 353-366. MR 47 #6507; Z 291.05009. W. Lea, Solution of the question 2244, Educational Times Reprints 9 (1868) 35-36. W. Lea, Solution of a question (2755), Educational Times Reprints 22 (1874) 74-76. J. Leech, A presentation of the Mathieu group M,,,Canad. Math. Bull. 12 (1964) 41-43. MR 41 #308; Z 175, p. 303. J . Leech and N.J.A. Sloane, New sphere packings in dimensions 9-15, Bull. Amer. Math. SOC.76 (1970) 1006-1010. MR 42 #965; Z 198, p. 551. J. Leech and N.J.A. Sloane, Sphere packings and error-correcting codes, Canad. J. Math. 23 (1971) 718-745. MR 44 #3211. F.W. Levi, Finite Geometrical Systems (Calcutta Univ. Press, 1942) MR 4, p. 49; Z 60, p. 323. R.J. Lewyckyj, see [C17]. S. Lin, see [H62]. C.C. Lindner, Construction of quasigroups using the singular direct product, Proc. Amer. Math. SOC.29 (1971) 263-266. MR 43 #6354; Z 217, p. 83. C.C. Lindner, Embedding partial idempotent Latin squares, J. Combinatorial Theory (A) 10 (1971) 240-245. MR 43 #1862. C.C. Lindner, Finite partial cyclic triple systems can be finitely embedded, Algebra Univ. 1 (1971) 93-96. MR 46 #1617; Z 219.05011. C.C. Lindner, Identities preserved by the singular direct product, Algebra Univ. 1 (1971) 86-89. MR 44 #5401; Z 221.20099. C.C. Lindner, Identities preserved by the singular direct product, 11, Algebra Univ. 2 (1972), 113-117. MR 46 #5519; Z 249.20038. C.C. Lindner, Finite embedding theorems for partial Latin squares, quasigroups and loops, J. Combinatorial Theory (A) 13 (1972) 339-345. MR 47 #3200; Z 246.05012. C.C. Lindner, Construction of nonisomorphic reverse Steiner quasigroups, Discrete Math. 7 (1974) 281-288. MR 48 #11379; Z 275. 20118. C.C. Lindner, A simple construction of disjoint and almost disjoint Steiner triple systems, J. Combinatorial Theory (A) 17 (1974) 204-209. Z 287.05021. C.C. Lindner, Construction of Steiner triple systems having exactly one triple in common, Canad. J. Math. 26 (1974) 225-232. MR 50 #1914; Z 287.05020. C.C. Lindner, On the construction of nonisomorphic Steiner quadruple systems, Colloq. Math. 29 (1974) 303-306. MR 54 #5006; Z 281.05010. C.C. Lindner, Some remarks on the Steiner triple systems associated with Steiner quadruple systems, Colloq. Math. 32 (1975) 301-306. MR 53 #5329; Z 274.05013; Z 306.05006. C.C. Lindner, On the structure of the Steiner triple systems derived from Steiner quadruple systems, Colloq. Math. 34 (1975) 137-142. MR 54 #2499; Z 321.05016.
334
L L20 1
[L32] [L331
[L34] [L35] 11-36]
11-37]
[L3X] [ 1.391
1.Doyen, A. Rosa C.C. Lindner. Construction of Steiner quadruple systems having large numbers of nonisomorphic associated Steiner triple systems, Proc. Amer. Math. SOC.49 (1975) 256-260. MR 52 #10448: Z 304.05006. C.C. Lindner, Disjoint finite partial triple systems can be embedded in disjoint finite Steiner triple systems, J. Combinatorial Theory (A) 18 (1975) 126-129. MR 50 #9615; Z 298.050 17. C.C. Lindner, A partial Steiner triple system of order n can be embedded in a Steiner triple system of order 6n +3. J. Combinatorial Theory (A) 18 (1975) 349-351. MR 52 #129: Z 304.05005. C.C. Lindner. A hrief up-to-date survey of finite embedding theorems for partial quasigroups. Prw. Conf. Algebraic Aspects of Combinatorin (Toronto 1975). Congressus Numerantium Xlll (Utilitas Math.. Winnipeg, 1975) 53-78. MR 52 #7929; Z 323.20074. C.C. Lindner, Embedding orthogonal partial Latin squares, Proc. Amer. Math. SOC.59 (1976) 184-186. MR 53 #12987; Z 342.05012. C.C. Lindner. Emhedding block designs into resolvable block designs, Ars Combinatoria 1 (1976) 215-219. MR 54 #2495; Z 333.05006. C.C. Lindner, Tuo finite embedding theorems for partial 3-quasigroups, Discrete Math. 16 (1976) 271-277. Z 352.20059. C.C. Ihdner. A note on disjoint Steiner quadruple systems, Ars Combinatoria 3 (1977) 271-276. MR 56 #5322. C.C. Lindner. Every Steiner quadruple system can be embedded in a resolvable Steiner quadruple system. Ars Combinatoria 3 (1977) 75-88. MR 56 #5321. C.C. Lindner. Embedding Steiner triple systems into Kirkman triple systems and Latin squares into Latin squares having orthogonal mates (to appear). C.C. Lindner and T. Evans, Finite Embedding Theorems for Partial Designs and Algebras. Seminaire de Mathematiques Supkrieures. UniversitP de Montreal (Les Presses dc I'Universite de Montreal, 1977) 196 pp. Z 363.05017. C.C. Lindner. E. Mendelsohn and A. Rosa, On the numher of I-factorizations of the complete graph, J. Combinatorial Theory (B) 20 (1976) 265-282. Z 293.05156; Z 321.0s 131 C.C. Lindner and N.S. Mendelsohn. Construction of perpendicular Steiner quasigroups, Aequat. Math. 9 11973) 150-156. MR 48 #6305; Z 267.20067. C.C. Lindner and K.T. Phelps, A note on partial parallel classes in Steiner systems, Discrete Math. 24 (1978) 109-1 12. C.C. Lindner and A. Rosa, Construction ,of large sets of almost disjoint Steiner triple systems, Canad. J . Math. 27 (1975) 256-260. MR 5 1 #5334: Z 321.05017. C.C. Lindner and A. Rosa, On the existence of automorphism free Steiner triple systems, J. Algehra 34 (1975) 430-143. MR 53 #2707: 2 308.05014. C.C. Lindner and A. Rosa. Finite embedding theorems for partial Steiner triple systems, Discrete Math. 13 (1975) 31-39. MR 52 #128; Z 309.05013, C.C. Lindner and A. Rosa, Steiner triple systems having a prescribed number of triples in common. Canad. J. Math. 27 (1975) 1166-1175. MR 56 #11819: Z 336.05008. Corrigendum: 30 (19781 896. C.C. Lindner and A. Rosa. Finite embedding theorems for partial triple systems with A > 1. Ars Combinatorial I (1976) 159-166. MR 53 #12973; Z 334.05021. C.C. Lindner and A. Rosa. Steiner quadruple systems all of whose dcrived Steiner triple systems are nonisomorphic. J. Combinatorial Theory (A) 21 (1976) 35-43. MR 53 #12974: 2 332.05013. C.C. Lindner and A. Rosa. There are at least 31,021 non-isomorphic Steiner quadruple systems of order 16, Utilitas Math. 10 (1976) 61-64. MR 54 #12539; Z 351.05012. C.C. Lindner and A: Rosa, A partial Room square can be embedded in a Room square, J. Combinatorial Theory ( A ) 22 (1977) 97-102. Z 349.0S014. C.C. Lindner and A. Rosa. Finite embedding theorems for partial Steiner quadruple systems. Bull. SOC. Math. Belg. 27 (1975) 315-323. C.C. I h d n e r and A. Rosa, Steiner quadruple systems-a survey, Discrete Math. 22 (1978) 147- 18 1.
Bibliography and survey of Steiner sysrems
335
C.C. Lindner and T.H. Straley, Construction of quasigroups containing a specified number of subquasigroups of a given order, Algebra Univ. 1 (1971) 238-247. MR 45 #3618; 2 229.20074. C.C. Lindner and T.H. Straley, A note on nonisomorphic reverse Steiner quasigroups, Publicationes Math. 23 (1976) 85-87. MR 54 #443. C.C. Lindner and T.H. Straley, A note on nonisomorphic Steiner quadruple systems, CMUC I 5 (1974) 69-74. MR 49 #79; 2 281.05011. C.C. Lindner, see CG16.11, [K25]. J.H.van Lint, see [C6], [D48]. P. Lorimer, A class of block designs having the same parameters as the designs of points and lines in a projective 3-space, Combinatorial Mathematics, Proc. 2nd Austral. Conf., Lecture Notes in Math. 403 (Springer, Berlin, 1974) 7 3 7 8 . MR 52 #130; 2 319.05010. E. Lucas, Rtcreations mathtmatiques, Vol. 2 (Gauthier-Villars, Paris, 1883) (Sixieme rkcrtation: Les jeux de demoiselles, pp. 161-197). H. Luneburg, Steinersche Tripelsysteme mit fahnentransitiver Kollineationsgruppe, Math. Ann. 149 (1963) 261-270. MR 26 #2933; 2 105, p. 131. H. Luneburg, Fahnenhomogene Quadrupelsysteme, Math. Z. 89 (1965) 82-90. MR 32 #1605; Z 135, p. 208. H. Luneburg, Uber die Gruppen von Mathieu, J. Algebra 19 (1968) 194-210. MR 38 #226; 2 169, p. 35. H. Luneburg, Transitive Erweiterungen endlicher Permutationsgruppen, Lecture Notes in Math. 84 (Springer, Berlin, 1969) MR 39 #2858; Z 217, p. 73. S.O. Macdonald, A characterization of affine triple systems, Ars Combinatoria 2 (1976) 23-24. S.O. Macdonald, Sum-free sets in loops, in: C.H.C. Little, ed., Combinatorial Mathematics V, Proc. 5th Australian Conf. 1976, Lecture Notes in Mathematics 622 (Springer, Berlin, 1977) 141-147. 2 337.05001. C.R. Maclnnes, Finite planes with less than eight points on a line, Amer. Math. Monthly 14 (1907) 171-174. F.J. MacWilliams, N.J.A. Sloane and J.G. Thompson, On the existence of a projective plane of order 10, J. Combinatorial Theory (A) 14 (1973) 66-78. MR 47 #1644; Z 251.05020. S. Magliveras, see [K37]. H.B. Mann, Analysis and Design of Experiments (Dover, New York, 1949) MR 11, p. 262. A.A. Markov, Ob odnoj kombinatornoj zadaEe, Problemy Kibernct. 15 (1965) 263-266. MR 35 #1497. E. Marsden, The school-girls’ problem, Knowledge 3 (1883) 183. D.R. Mason, On the construction of the Steiner system S(5, 8,24), J. Algebra 47 (1977) 77-79. MR 56 #3100. R. Mathon and A. Rosa, A census of Mendelsohn triple systems of order nine. Ars Combinatoria 4 (1977) 309-315. rM9.13 R. Mathon and S.A. Vanstone, On the existence of doubly resolvable Kirkman systems and equidistant permutation arrays, Discrete Math. (to appear). R. Mathon, see [G3.1], [G8], [G9], [GlO]. H.F. Mattson Jr., see [AIS], [A16], [A17], [A18], [A19], [A20], [A21], [A22], [A23], [A24]. T. Matzner, see [H41]. J. McKay, see [J8]. M. McLeish, On the number of conjugates of ternary quasigroups. Ph.D. Thesis, McMaster University, Hamilton, April 1977. E. Mendelsohn, On the groups of automorphisms of Steiner triple and quadruple systems, Proc. Conf. Algebraic Aspects of Combinatorics (Toronto 1975), Congressus Numerantium XI11 (Utilitas Math., Winnipeg, 1975) 255-264. MR 52 #5438; Z 322.05012. E. Mendelsohn, The smallest non-derived Steiner triple system is simple as a loop, Algebra Univ. 8 (1978) 256-259. MR 57 #2934; Z 375.20059. E. Mendelsohn, On the groups of automorphisms of Steiner triple and quadruple systems, J. Combinatorial Theory (A) 25 (1978) 97-104.
336
J. Doyen, A. Rosa
[M13.1] E. Mendelsohn, Every (finite) group is the group of automorphisms of a (finite) strongly regular graph, Ars Combinatoria 6 (1978) 75-86. E. Mendelsohn, see [A10.2], [L31]. N.S. Mendelsohn, Combinatorial designs as models of universal algebras, Recent Progress in Combinatorics, Proc. 3rd Waterloo Conf. 1968 (1969) 123-132. MR 41 #85; Z 192, p. 333. N.S. Mendelsohn. A theorem on Steiner systems, Canadian J. Math. 22 (1970) 1010-1015. MR 42 #1677. Z 192. pp. 333; Z 206, p. 21. N.S. Mendelsohn, A theorem on Steiner systems, Canad. J . Math. 22 (1970) 1010-1015. #15X7; Z 212, p. 35. N.S.Mendelsohn, A natural generalization of Steiner triple systems, Computers in Number Theory (Academic Press, New York, 1971) 323-338. MR 48 #122; Z 216, p. 301. N.S. Mendelsohn, A single groupoid identity for Steiner loops, Aequat. Math. 6 (1971) 228-230. MR 45 #6969; 2 244.20087. N.S. Mendelsohn, Intersection numbers of [-designs, Studies in Pure Math., papers presented to Richard Rado o n the Occasion of his sixty-fifth Birthday (1971) 145-150. MR 42 #5819; Z 222.05018. N.S. Mendelsohn and S.H.Y. Hung, On the Steiner systems S(3.4. 14) and S(4, 5, 15), Utilitas Math. 1 (1972) 5-95. MR 46 #1618; Z 258.05017. N.S. Mendelsohn, see [B29], [B30], [H60], [J5], [L32], ES431. A.F.H. Mertelsmann, Das Problem der 15 Pensionatdamen, Zeitschr. Math. Phys. 43 (1898) 329-3 34. D.M. Mesner. see [K34], [K35], [K36], [K37]. R. Metz and H. Werner, On automorphism groups of finite Steiner triple systems, Teorie Combinatorie 11, Colloq. Internat. Roma 1973, Atti dei Convegni Lincei 17 (Roma. 1976) 373-376. MR 57 #12253. R. Metz. see LG3.21. IM22.11 G.L. Miller. On the nlognisomorphism technique, Proc. Tenth Annual ACM Sympos. Theory of Computing, San Diego, CA (May 1978) 51-58. [M23] W.H. Mills, On the covering of pairs by quadruples 1, J. Combinatorial Theory (A) 13 (1972) 55-78. MR 45 #8534; Z 243.05024. [M24] W.H. Mills. On the covering of pairs by quadruples 11, J. Combinatorial Theory (A) 15 (1973) 138-166. MR 47 #8316; 2 261.05022. [M25] W.H. Mills. Covering problems, Proc. 4th South-Eastern Conf. Combinatorics, Graph Theory and Computing (Boca Raton, 1973). Congressus Numerantium VIII (Utilitas Math., Winnipeg, 1973) 23-52. MR 50 #12771; Z 318.05003. [M25.1] W.H. Mills, A new 5-design, Ars Combinatoria 6 (1978) 193-195. A. Mitschke and H. Wernir, On groupoids representable by vector spaces over finite fields, Arch. Math. 24 (1973) 14-20. MR 47 #5148; 2 255.20049. E.H. Moore, Concerning triple systems, Math. Ann. 43 (1893) 271-285. E.H. Moore. The group of holoedric transformation into itself of a given group, Bull. Amer. Math. SOC.1 ( 1 8 9 4 9 5 ) 61-66. E.H. Moore. Concerning triple systems, Rend. Circ. Mat. Palermo 9 (1895) 86. E.H. Moore. Tactical memoranda 1-111, Amer. J. Math. 18 (1896) 264-303. E.H. Moore, Concerning regular triple systems, Bull. Amer. Math. Soc. 4 (1897) 11-16. E.H. Moore, Concerning abelian-regular transitive triple systems, Math. Ann. SO ( 1898) 225-240. E.H. Moore, Concerning the general equations of the seventh and eighth degrees, Math. Ann. 5 1 (1899) 417-444. I-. Mouette. Groupemenis en triades. Probleme de Kirkman gknkralis6, Mathesis 65 (1956) 46-52. MR 17, p. 1173; Z 71, p. 13. L.. Mouette, Rrcherches sur la thiorie des triades, Mathesis 66 (1957) 283-287. MR 20 #5743; 2 79, p. 10. P. Mulder, Kirkman-systemen, Academisch Proefschrift ter verkrijging van den graad van doctor in de Wis-en Natuurkunde aan de Rijksuniversiteit te Groningen, Leiden 1917. R.C. Mullin and C.C. Lindner, Lower bounds for maximal partial parallel classes in Steiner systems, J. Combinatorial Theory (A) 26 (1979) 314-318.
Bibliography and suruey of Steiner systems
“11
“51
“81
“91
337
R.C. Mullin and E. Nemeth, On furnishing Room squares, J. Combinatorial Theory 7 (1969) 266-272. MR 41 #5228; Z 186, p. 19. R.C. Mullin and E. Nemeth, On the nonexistence of orthogonal Steiner systems of order 9, Canad. Math. Bull. 13 (1970) 131-134. MR 41 #3297; Z 195, p. 31. R.C. Mullin and A. Rosa, Orthogonal Steiner systems and generalized Room Squares, Proc. 6th Manitoba Conf. Numerical Math. (Winnipeg 1976), Congressus Numerantiurn XVIII (Utilitas Math., Winnipeg, 1977) 315-323. R.C. Mullin, see [H53], [S42], [S44], [S45]. C.St.J.A. Nash-Williams, Simple constructions for balanced incomplete block designs with block size three, J. Combinatorial Theory (A) 13 (1972) 1-6, MR 46 #3337; Z 258.05012. E. Nemeth, see [D28], [D28.1], [M38], [M39]. G.L. Nemhauser, see [F21]. I.P. Neporoinev, Novoje dokazatel’stvo nevozmoinosti projektivnoi ploskosti s 43 tofkami, Wen. Zap. Permsk. Gos. Pedagog. Inst. 20 (1959) 33-38. I.P. Neporoinev and A. Ja. Petrenjuk, Konstruktivnoje perecislenije sistem grupp par i oglavlennyje sistemy trojek Stejnera, Kombinatornyj Analiz 2 (1972) 17-37. 1.P. Neporoinev and A.Ja. Petrenjuk, Konstruktivnoje perecislenije sistem grupp par i oglavlennyje sistemy trojek Stejnera (prodolienije), Kombinatornyj Analiz 3 (Izdatel’stvo MGU, Moskva, 1974) 28-42. E. Netto, Die Substitutionentheorie und ihre Anwendungen in der Algebra (Teubner, Leipzig, 1882). (English translation by F. N. Cole: The Theory of Substitutions and its Applications to Algebra (A.A. Register Publ. Co., 1892; reprinted by Chelsea, New York. 1964).) Chap. XII: Equations with rational relations between three roots. E. Netto, Zur Theorie der Tripelsysteme, Math. Ann. 42 (1893) 143-152. E. Netto, Kombinatorik, Encyklopadie der mathematischen Wissenschaften Band I, Heft 1, 28-46. (Teubner, Leipzig, 1898). E. Netto, Lehrbuch der Combinatorik (Teubner, Leipzig, 1901) 2nd ed. (with notes by V. Brun and Th. Skolem) 1927; reprinted by Chelsea, New York, 1968. MR 20 #1632. D.J. Newman, see [K17]. S. Niven, On the number of k-tuples in maximal systems m(k, I, n), Combinat. Structures and their Applications (Proc. Conf. Calgary 1969) (Gordon and Breach, New York, 1970) 303-306. MR 42 #4412; Z 278.05025. R. Noda, Steiner systems which admit block transitive automorphism groups of small rank, Math. Z. 125 (1972) 113-121. MR 45 #85; Z 218.05007. M. Noether, Uber die Gleichungen achten Grades und ihr Auftreten in der Theorie der Curven vierter Ordnung. Math. Ann. 1.5 (1879) 80-1 10 M. Noether, Note uber die Siebensysteme von Kegelschnitten, welche durch die Beriihrungspunkte der Doppeltangenten einer ebenen Curve vierter Ordnung gehen, Math. Ann. 46 (1895) 545-556. J. Novak, Piispsvek k teorii kombinaci, Casopis Pe‘st. Mat. 88 (1963) 129-141. MR 28 #2977; Z 128, p. 244. J. NovBk, Ueber gewisse minimale Systeme von k-Tupeln, CMUC 11 (1970) 397-401. MR 42 #4413; Z 206, p. 293. J. Novik, MaximBlni systCmy trojic z 12 prvki, Mathematics (Geometry and Graph Theory) (Univ. Karlova, Praha, 1970) 105-110. MR 43 #3141; 2216, p. 302. J. NovBk, Uber gewisse Tripel-Systeme, Combinat. Theory and its Applications (Balatonfured, Hungary 1969). 111, Colloq. Math. SOC.J. Bolyai 4 (North Holland, Amsterdam, 1970) 821-828. MR 45 #8543; Z 217, p. 18. J. Novhk, Edge bases of complete uniform hypergraphs, Mat. cas. 24 (1974) 43-57. MR 59 #9710; Z 273.05132. J. NovBk, A note on disjoint cyclic Steiner triple systems, Recent Advances in Graph Theory (Proc. Symp. Prague 1974) (Academia, Praha 1975) 439-440. MR 52 #2920; Z 365.05012. J. Novak and J. Vild, Ntktert maximahi systtmy trojic, Sbornik ve‘deckfch praciVys. s’koly strojni a textilni v Liberci (1966) 29-36.
338 “19.11
J . Doyen, A. Rosa
J. Novak. On certain minimal systems of k-tuples. Problemes Conbinatoires et Thiorie des Graphes. Orsay 1976, Colloq. Internat. C.N.R.S. No. 260 (Editions du CNRS, Paris, 1978) 3 17-320. J. Novak, see [B36]. W. Oberschelp, Lotto-Garantiesysteme und Blockplane, Math.-Phys. Semesterber. 19 (1972) 55-67. MR 51 #7901. E.S. O’Keefe. Verification of a conjecture of Th.Skolem, Math. Scand. 9 (1961) 80-82. MR 23A #2331; Z 105, p. 250. B. d’Orgeval, Sur certains 2-systemes et le probleme de Kirkman, Acad. Roy. Belg. Bull. CI. Sci. 53 (1967) 21-25. MR 35 #6020; Z 152, p. 190. C.D. OShaughnessy, A Room design of order 14, Can. Math. Bull. 11 (1968) 191-194. MR 36 #3940. C.D. OShaughnessy, On Room squares of order 6 m + 2 , J. Combinatorial Theory (A) 13 (1972) 306-314. MR 46 #1619; Z 246.05018. E.R. Ott, Finite projective geometries, PG(k, p”), Amer. Math. Monthly 44 (1937) 86-92. Z 16, p. 37. L.J. Paige, A note on the Mathieu groups, Canad. J. Math. 9 (1957) 15-18. MR 18, p. 871; Z 77, p.32. E.T. Parker, On sets of pairwise disjoint blocks in Steiner triple systems, J. Combinatorial Theory ( A) 19 (1975) 113-114. MR 51 #12559; Z 302.05012. E.T. Parker, see [B54]. B. Peirce, Cyclic solutions of the school-girl puzzle. Astronomical Journal (U.S.A.) 6 (1860) 169- 174. N.R. Pekelharing, Over het combinatorisch probleem van Steiner, Academisch Proefschrift ter verkrijging van den graad van doctor in de Wis- en Natuurkunde aan de RijksUniversiteit re Groningen, Groningen, 1918. N.R. Pekelharing, Stelsels van 7 vier-deelingen van 8 elementen, Nieuw Arch. Wisk. (2) 13 (1921) 361-382. J. Pelikhn. Properties of balanced incomplete block designs, Combinat. Theory and its Applications (Balatonfured, Hungary 1969). 111, Colloq. Math. Soc. J. Bolyai 4 (North Holland, Amsterdam.1970) 869-889. MR 46 #8859; Z 217, p.20. J. Pelikhn, see [G4]. R. Peltesohn. Das Turnierproblem fur Spiele zu je dreien, Dissertation Berlin, August Pries, Leipzig, 1936. Z 13, p. 338. R. Peltesohn. Eine Losung der beiden Heffterschen Differenzenprobleme. Compositio Math. 6 (1939) 251-267. Z 20, p. 39. P. Penning, see [B25.1]. A.Ja. Pctrcnjuk. Nekotoryje ocenki Eisel Mano, Vestnik Mosk. Univ. Mat. Meh. 25 (1970) 31-35. (English translation: Some bounds for Mano numbers, Vestnik Mosk. Univ. 25 (1970) 87-90.) MR 44 #2634. A.Ja. Petrenjuk, 0 neizomorfnych oglavlennych sistemach trojek ktejnera i sistemach grupp par, Kibernetika (Kiev) No. 4 (1971) 93-102. Z 226.05014. A.Ja. Petrenjuk. Priznaki neizornorfnosti sistem trojek Stejnera, Ukr. Mat. Z. 24 (1972) 772-780. 86 1. (English translation: Tests for nonisomorphic Steiner triple systems. Ukr. Mat. Zh. 24 (1972) 620-626.) MR 48 #123: Z 256.05007. A.Ja. Petrenjuk, see “31, “41, [PIZ]. L.P. Petrenjuk and A.Ja. Petrenjuk, Postrojenie nekotorych klassov kubireskich grafov i neizomorfnost’ Kirkmanovych sistem trojek, Kombinatornyj Analiz 4 (Izdat. Mosk. Univ.. Moskva, 1976) 73-77. MR 56 #15499. K.T. Phelps. Derived triple systems of order 15, MSc. Thesis, Auburn University, Alabama 1975. 56 pp. K.T. Phelps, Some sufficient conditions for a Steiner triple system to be a derived triple system, J. Combinatorial Theory ( A ) 20 (1976) 393-397. MR 53 #12975; Z 329.05010. K.T. Phelps, Some derived Steiner triple systems, Discrete Math. 16 (1976) 343-352. MR 54 #l0041; Z 351.05011. K.T. Phelps, Rotational Steiner quadruple systems, Ars Combinatoria 4 (1977) 177-185, MK 57 #12254.
Bibliography and survey of Steiner systems
[ROI [R0.1]
339
K.T. Phelps, A construction of disjoint Steiner quadruple systems, Proc. Eighth S.-E. Conf. Combinatorics, Graph Theory and Computing, Baton Rouge 1977, Congressus Numerantium XIX (Utilitas Math., Winnipeg, 1977) 559-567. K.T. Phelps, A note on the construction of cyclic quadruple systems, Colloq. Math. (to appear). K.T. Phelps, Automorphism free latin square graphs (to appear). K.T. Phelps, A note on partial S(s - 1, s + 1. u) subsystems of a S(s, s + 1, u), Roc. Ninth S.-E. Conf. Combinatorics, Graph Theory and Computing (Boca Raton, 1978) Congressus Numerantiurn XXI (Utilitas Math., Winnipeg 1978) 581-586. K.T. Phelps, Latin square graphs and their automorphism groups, Ars Combinatoria 7 (1979) 273-299. K.T. Phelps, Infinite classes of cyclic Steiner quadruple systems (to appear). K.T. Phelps, see [L33]. G. Pickert, Projektive Ebenen, Springer, Berlin 1955. Kap. 12: Endliche Ebenen. MR 17, p. 399; Z 66, p. 387. 2nd ed (Springer, Berlin, 1975) MR 51 #6577; Z 307.50001. G. Pickert, Plans de blocs, Bull. Soc. Math. Belg. 24 (1972) 43-55. MR 52 #13394; Z 247.05002. W.A. Pierce, The impossibility of Fano’s configuration in a projective plane with eight points per line, Proc. Amer. Math. SOC.4 (1953) 908-912. MR 15 p. 460; Z 52, p. 165. W.-L. Piotrowski, see [K23]. J. Power, On the problem of the fifteen school girls, Quart. J. Pure Appl. Math. 8 (1867) 236-25 1. E.J.F. Primrose, Kirkman’s schoolgirls in modern dress, Math. Gaz. 60 (1976) 292-293. Z 343.05004. K. Pukanow, On the number of non-isomorphic configurations C(4,2,1, u), Prace Naukowe Instytutu Matematyki i Fizyki Teoretycznej Politechniki Wroclawskiej, Nr. 9 (1973) 45-47. MR 50 #1918; Z 278.05026. K. Pukanow, On number of nonisomorphic block systems B(4,1, u), Prace Naukowe Instytutu Matematyki Politechniki Wroclawskiej, No. 12 Analyza Kombinatoryczna, Wroclaw (1976) 11-37. Z 343.05012. N. K. Pukharev, Nekotoryje svojstva regul’arnych konefnych ploskostej LobaEevskogo, UEen. Zap. Permsk. Gos. Univ. Mat. 103 (1Y63) 228-230. MR 31 #659. N.K. Pukharev, Ob A:-algebrach i regul’arnych konetnych ploskostjach, Sib. Mat. Z. 6 (1965) 892-899. MR 32 #8249; Z 148, p. 408. N.K. Pukharev, 0 postrojenii A:-algebr, Sib. Mat. Z. 7 (1966), 724-727. (English translation: Construction of A:-algebras, Sib. Math. J. 7 (1966) 577-579.) MR 33 #7450. N.K. Pukharev, Postrojenie otdel’nych klassov regul’arnych konechnych prostranstv, Izv. VysS. Ufebn. Zaved. Matematika 69 (1968) 85-88. MR #7017; Z 155, p. 490. T.M. Putnam, Concerning quadruple systems, Bull. Amer. Math. SOC.8 (1901-02) 434. R.W. Quackenbush, Near-Boolean algebras I: Combinatorial aspects, Discrete Math. 10 (1974) 301-308. MR 5 1 #314; Z 326.05015. R.W. Quackenbush, Near vector spaces over GF(q) and ( u , q + l,l)-BIBDS, Linear Alg. Appl. 10 (1975) 259-266. MR 51 #5335; Z 343.05013. R.W. Quackenbush, Algebraic aspects of Steiner quadruple systems, Proc. Conf. Algebraic Aspects of Combinatorin (Toronto, 1975). Congressus Numerantium XI11 (Utilitas Math.,Winnipeg, 1975) 265-268. MR 52 #5431; Z 331.05009. R.W. Quackenbush, Varieties of Steiner loops and Steiner quasigroups, Canad. J. Math. 28 (1976) 1187-1198. MR 54 #12946; Z 359.20070. P. Quattrocchi, Un metodo per la costruzione di certe strutture che generaliuano la nozione di piano grafico, Le Matematiche 21 (1966) 377-386. MR 34 #2482; Z 147, p. 199. F. Rado, On semi-symmetric quasigroups, Aequat. Math. 11 (1974) 250-255. M. Raghavachari, Some properties of the width of the incidence matrices of balanced incomplete block designs and Steiner triple systems, Sankhya (B) 39 (1977) 181-188. M. Rahman, see [B43]. B.L. Raktoe, see [H36]. C.R. Rao, A study of BIB designs with replications 11 to 15, Sankhya Ser. A 23 (1961) 117-127. MR 25 #732; Z 99 p. 354.
1. Doyen, A. Rosa
D.K. Ray-Chaudhuri, On some connections between graph theory and experimental designs and some recent existence results, Graph Theory Appl., Proc. Advanced Sem. (Madison, WI, 1969), 1970. 149-166. MR 42 #115; Z 225.05119. D.K. Ray-Chaudhuri. Recent developments on combinatorial designs, Actes Congr. Internat. Math. (Nice 1970), T. 3 (Gauthier-Villars, Paris, 1971) 223-227. MR 54 #10042; Z 226.050 11. D.K. Ray-Chaudhuri and R.M. Wilson, O n the existence of resolvable balanced incomplete block designs, Combinat. Structures and their Applications (Proc. Conf. Calgary 1969) (Gordon and Breach, New York 1970) 331-341. M R 42 #1678; Z 265.05004. D.K. Ray-Chaudhuri and R.M. Wilson, Solution of Kirkman’s school-girl problem, Proc. Symp. Pure Math. 19 (Amer. Math. SOC..Providence, RI. 1971) 187-203. M R 47 #3195; Z 248.05009. D.K. Ray-Chaudhuri and R.M. Wilson, The existence of resolvable block designs. A Survey of Combinatorial Theory (North Holland, Amsterdam, 1973) Chap. 30. 361-375. MR 50 #12761; Z 274.05010. D.K. Ray-Chaudhuri. see [H30]. M.Reiss. Uber eine Steinersche combinatorische Aufgabe. welche im 3Ssten Bande dieses Journals, Seite 181, gestellt worden ist, J. reine angew. Math. 56 (1859) 326-344. A. Renyi. see [ElO]. M. Richardson. see [HW]. G. Ringel. Uber das problem der Nachbargebiete auf orientierbaren Flachen, Abh. Math. Sem. Univ. Hamburg 25 (1961) 105-127. MR 23A #2876: Z 182. p. 266. G. Ringel. Die toroidale Dicke des vollstandigen Graphen. Math. Z. 87 (196.5) 19-26. MR 30 #2489; Z 132, p. 213. R.M. Robinson, Triple systems with prescribed subsystems. Notices Amer. Math. SOC. 18 (1971) 637. R.M. Robinson, The structure of certain triple systems. Math. Comput. 29 (1975) 223-241. MR 52 #5440: Z 293.05015. B. Rokowska. Some new constructions of 4-tuple systems, Colloq. Math. 17 (1967) 11 1-121. MR 35 #4117; 2 158. p. 14. B. Rokowska, Some remarks on the number of different triple systems of Steiner, Colloq. Math. 22 (1971) 317-324. M R 44 #3890; 2 218.05008. B. Rokowska, 0 nowych konfiguracjach C(k,1, n , A ) , Prace Naukowe Instytutu Matematyki i Fizyki Teoretycznej Politechniki Wroclawskiej Nr. 4 (1971) 61-64. M R 49 #8882. 8 . Rokowska. O n the number of different triple systems of Steiner. Prace Naukowe Instytutu Matematyki I Firyki Teoretycznej Politechniki Wroclawskiej N r . 6 (1972) 41-57. MR SO #6885; Z 2S3.05005. B. Rokowska. On the number of non-isomorphic Steiner triples. Colloq. Math. 27 (1973) 149-160. MR 48 #3759; Z 254.05006. B. Rokowska. A new construction of the block systems B(4. 1.25) and B(4, 1,28), Colloq. Math. 38 (lY77) 165-167. M R 57 #143; 2 377.05005. [R17.1] B. Rokowska. Non-isomorphic Steiner triples with subsystems, Colloq. Math. 38 (1977) 152-164. MK 57 #142: Z 377.05004. [RI 7.21 B. Rokowska, On resolvable quadruple systems, Instytut Matematyki Politechniki Wroclawskiej. Komunikat Nr. 99 (1977) 17 pp. [R18] B. Rokowska and M. Wojtas, Nonisomorphic balanced incomplete block designs B(v, I , 5). Graphs, Hypergraphs, Block Designs: Sci. Pap. Inst. Math. Wroclaw Techn. IJniv. 17. Stud. RCS. 13 (1977, 19-24. 2 378.05012. [ R l 9 ] A . Rosa. Ispol’zovanie grafov dlja reienija zadafi Kirkmana. Mat. -Fyz. e a s . 13 (1963) 105-113. MR 28 #1615; Z 145, p. 441. LR201 A. Rosa. Zametka k odnoj zadaEe PO teorii grafov. Mat. -Fyz. c a s . 13 (1963) 238-239. M R 32 #2349: Z 145. p. 441. A. Rosa. Poznhmka o cyklickych Steinerov9ch systemoch trojic, Mat. -Fyz. Cas. 16 (1966) [Rl] 285-290. M R 35 #2759; Z 147, p. 267. I R22] A. Rosa, On the chromatic number of Steiner triple systems. Combinat. Structures and their Applications (Proc. Conf. Calgary 1969) (Gordon and Breach, New York, 1970) 369-371. MR 43 #6110; Z 264.05012.
Bibliography and suruey of Steiner systems
34 1
A. Rosa, Steiner triple systems and their chromatic number, Acta Fac. Rerum Natur. Univ. Comen. Math. 24 (1970) 159-174. MR 45 #87; Z 213, p. 262. [R24] A. Rosa, Subsystems of Steiner systems, Ann. N.Y. Acad. Sci. 175 (1970) 327-328. MR 45 #3219; Z 242.05017. [R25] A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972) 61-71. MR 45 #4996; Z 242.05016. [R26] A. Rosa, On the falsity of a conjecture on orthogonal Steiner triple systems, J. Combinatorial Theory (A) 16 (1974) 126128. MR 48 #10845; Z 277.05008. [R27] A. Rosa, A theorem on the maximum number of disjoint Steiner triple systems, J. Combinatorial Theory (A) 18 (1975) 305-312. MR 51 #191; Z 306.05008. [R28] A. Rosa, Algebraic properties of designs and recursive constructions, Proc. Conf. Algebraic Aspects of Combinatorics (Toronto,l975), Congressus Numerantium XI11 (Utilitas Math., Winnipeg, 1975) 183-200. MR 56 #8391; Z 328.05012. A. Rosa, see [D44], [D44.1], [G3.1], [G22], [K25], [K26], [L31]. [L34], [L35], [L36], [L37], LI-381, [L391, [L401, W11, D-421, [L431, [M91, [M401. [R29] D.P. Roselle, Distributions of integers into s-tuples with given differences, Conf. Numerical Methods (Winnipeg, 1971) 31-42. MR 49 #211; Z 267.05018. [R29.1] R. Roth, The catalogue of Hall triple systems of small orders, Notices Amer. Math. SOC.26 (1979), No. 1, Abstract 763-17-5;p. A54. A. Rudvalis, see [H61]. B.T. Rumov, 0 postrojenii odnoj beskonetnoj serii sistem trojek Kirkmana, Voprosy Kibernetiki 16 (1975) (Moskva, 1975) 77-78. A.K. Rybnikov, see [R31]. N.M. Rybnikova and A.K. Rybnikov, Novoje dokazatel'stvo nesuSEestvovanija projektivnoj ploskosti porjadka 6, Vestnik Mosk. Univ. Ser. I. Mat. Meh. 21 (1966) 20-24. MR 34 #4982; Z 149, p. 386. H.J. Ryser, Combinatorial Mathematics (Carus Math. Monograph No.14) (Wiley, New York, 1963) MR 27 #51; Z 112, p. 248. (Russian translation: Kombinatornaja matematika, Mir, Moskva 1966. Z 146, p. 11. French translation: Mathkmatiques combinatoires (Dunod, Paris, 1969) Z 165, p. 327.) H.J. Ryser, see [B61]. A. Sade, Quasigroupes demi-symbtriques, Ann. Soc. Sci. Bruxelles Ser. I 7 9 (1965) 133143. MR 34 #2760; Z 132, p. 265. S.H. Safford, Discussion of a problem proposed by Dr. Oswald Veblen, Amer. Math. Monthly 14 (1907) 84-86. A. Sainte-Lague, Avec des nombres et des lignes (Rbcrtations mathbmatiques) (Vuibert, Paris, 2" ed., 1943) Deuxieme partie, Chap. 5: Les triades, pp. 197-209. A. Samardiiski, 3-(u, 4, 1) block designs and 3-quasigroups (Macedonian), Bull. SOC.Math. Phys. Mactdoine 26 (1975-76) 15-18. MR 57 #5782; 2 381.05007. N. Sauer and J. Schonheim, Maximal subsets of a given set having no triple in common with a Steiner triple system on the set, Canad. Math. Bull. 12 (1969) 777-778. MR 40 #5456; 2 194, p. 10. S.R. Savur, A note on the arrangement of incomplete blocks, when k = 3 and A = 1, Ann. Eugenics 9 (1939) 45-49. F. Scheid, A tournament problem, Amer. Math. Monthly 67 (1960) 39-41. MR 23A #72; 2 93, p. 15. A. Schleiermacher, see [K4]. J. Schonheim, On coverings, Pacific J. Math. 14 (1964) 1405-1411. MR 30 #1954; Z 128, p. 245. J. Schonheim, On maximal systems of k-tuples, Studia Sci. Math. Hung. 1 (1966) 363-368. MR 34 #2485; Z 146, p. 14. J. Schonheim, On the number of mutually disjoint triples in Steiner systems and related maximal packing and minimal covering systems, Recent Progress in Combinatorics (Proc. 3rd Waterloo Conf. 1968) (Academic Press, New York, 1969) 311-318. MR 40 #5457; Z 196, p. 26. J. Schonheim, Semilinear codes and some combinatorial applications of them, Inform. Control 15 (1969) 61-66. MR 40 #8494; Z 192, p. 562.
[R23]
342
J. Doyen, A. Rosa
J. Schonheim, Partition of the edges of the directed complete graph into 4-cycles, Discrete Math. 11 (1975) 67-70. MR 51 #251. J. Schonheim, see [S4]. IS121 S. Schreiber, Covering all triples on n marks by disjoint Steiner systems, J. Combinatorial Theory (A) 15 (1973) 347-350. MR 48 #10846; Z 274.05021. [S13] S. Schreiber. Some balanced complete block designst Israel J. Math. 18 (1974) 31-37. MR 50 #141; Z 292.05004. H. Schubert, Zwolf Geduldspielen (Berlin, 1895) 20-25. H. Schubert, Mathematische Mussestunden (Goschen, Leipzig, 1898) vol. 2, pp. 49-66. 6 AuR. (Neubearb. v. F. Fitting), Walter de Gruyter, Berlin 1940. Z 22, p. 297. 10 Aufl. Walter de Gruyter, Berlin. 1943. 2 28, p. 196. R.-H. Schulz. Zur Geometrie der PSU (3, 4’): eine Klasse von Steinerschen Systemen. Geometriae Dedicata 3 (1974) 11-19. MR 49 #4811; Z 285.05024. B. Segre. Forme e geometrie hermitiane con particolare riguardo al caso finito. Ann. Mat. Pura Appl. (4) 70 (1965) 1-202. MR 35 #4802; Z 146. p. 167. B. Segre, lstituzioni di Geometria Superiore (Anno accad. 1963-64). Vol 3: Complessi, reti, disegni. lstituto Matematico “G. Castelnuovo”, Universita di Roma 1965. MR 37 #4700c. J.J. Seidel. see [B78], [B79], [G12], [G13]. E. Seiden. A method of construction of resolvable BIBD. Sankhya Ser. A 25 (1963) 393-394. MR 30 #2632: 2 124, p. 108. E. Seiden. see [A6]. N.V. Semakov and B.A. Zinoviev, SoverSennyje i kvazisoverSennyje ravnovesnyje kody, Problemy Peredafi lnformacii 5 (1969). Vyp. 2. 14-18, N.V. Semakov. see [ZO]. M.F. Semionov, see [A7]. S.S. Shrikhande. see [BS2]. [B53]. [B54]. W. Sierpinski. Sur un probleme de triades, C. R. SOC. Sci. Varsovie 33-38 (1946) 13-16. MR [S21] 8. p. 570: Z 60. p. 130. [S22] G.P. Sillitto. Note on Takeuchi’s table of difference sets generating balanced incomplete block designs. Rev. Inst. Internat. Statist. 32 (1964) 251. MR 30 #4362; 2 124, p. 108. C.C. Sims, see [H45]. S.A. Sirokova. Blok-schemy, Uspechi Mat. Nauk 23 (196X), vyp. 5 (143) 51-98. MR 38 #5649: Z 177. p. 227. (English translation: Block designs, Russian Mat. Surveys 23 (1968), NO. 5 . 47-94. Z 187, p. 162). Tb. Skolem. O m e n del kombinatoriske problemer, Norsk Mat. Tidsskr. 9 (1927) 87-105. Th. Skolem. Uhrr einige besondere Tripelsysteme mit Anwendung aul’ die Reproduktion gewisscr. Ouadrat-summen bei Multiplikation. Norsk. Mat. Tidsskr. 13 (1931) 41-51. Z 3. p. 145. [S261 Th. Skolem. O n certain distributions of integers in pairs with given differences, Math. Scand. 5 (1957) 57-68. MR 19, p. 1159; Z 84, p. 43. ~ 2 7 1 Th. Skolem, Some remarks o n the triple systems of Steiner. Math. Scand. 6 (1958) 273-280. MR 21 #5582; 2105, p. 250. L.A. Skornjakov. Projektivnyje ploskosti, Uspechi Mat. Nauk 6 (1951), vyp. 6 (46) 112-154. [S28] MR 13. p. 767; Z 45, p. 99. (English translation: Projective planes, Amer. Math. SOC. Translation No. 99, 58 pp. (1953). MR 15, p. 550. or: Translations Amer. Math. SOC.Ser. 1. l ( 1 9 6 2 ) 51-107.) N.J.A. Sloane. see [LS]. [L6]. [M4]. D. Smith, see [C13]. P. Smith. A doubly divisible nearly Kirkman system. Discrete Math. 18 (1977) 93-96. B. Sobocinski, A theorem of Sierpiriski on triads and the axiom of choice, Notre Dame J. Formal Logic 5 (1964) 51-58. MR 31 #3343; 2 134, p. 248. A. Sohczyk, Multi-functions associated with Steiner systems. Notices Amer. Math. SOC. 14 (1967) 925-926. J. Spencer, Maximal consistent families of triples. J. Combinatorial Theory 5 (1968) 1-8. MR 37 #2616: Z 157, p. 32.
Bibliography and suruey of Sreiner systems
343
G. Spoar, The connection of block designs with finite Bolyai-Lobachevsky planes, Math. Mag. 47 (1973) 101-102. Z 264.50012. W. Spottiswoode, On a problem in combinatorial analysis, London, Edinburgh and Dublin Philos. Mag. and J. Sci. (4) 3 (1852) 348-354. D.A. Sprott, A note on balanced incomplete block designs, Canad. J. Math. 6 (1954) 341-346. MR 15, p. 926; Z 55, p. 377. D.A. Sprott, Balanced incomplete block designs and tactical configurations, Ann. Math. Statist. 26 (1955) 752-758. MR 17, p. 572; Z 66, p. 127. D.A. Sprott, Some series of balanced incomplete block designs, Sankhya 17 (1956) 185-192. MR 18, p. 459; Z 72, p. 368. D.A. Sprott, Listing of BIB designs from r = 16 to r = 2 0 , Sankhya Ser. A 24 (1962) 203-204. MR 26 #1978; Z 105, p. 121. R. St. Andre, see [C13]. R.G. Stanton, The Mathieu groups, Canad. J. Math. 3 (1951) 164-174. MR 12, p. 672; Z 42, p. 256. R.G. Stanton and J.G. Kalbfleisch, The A - p problem: A = 1 and p = 3 , Combinatorial Mathematics and its Applications (Proc. 2nd Chapel Hill Conf.), Univ. of N.C. (May 1970) 451-462. MR 42 #2954; Z 213, p. 28. R.G. Stanton and J.G. Kalbfleisch, Coverings of pairs by k-sets, Ann. N. Y. Acad. Sci. 175 (1970) 366-369. MR 42 #87; Z 227.05014. R.G. Stanton, J.G. Kalbfleisch and R.C. Mullin, Covering and packing designs, Combinatorial Mathematics and its Applications (Proc. 2nd Chapel Hill Conf.), Univ. of N.C. (May 1970) 428-450. MR 42 #4424; Z 213, p. 28. R.G. Stanton and N.S. Mendelsohn, Some results on ordered quadruple systems, Proc. Louisiana Conf. Combinatorics, Graph Theory and Computing (Baton Rouge, 1970) 297309. MR 42 #5827; Z 229.05020. R.G. Stanton and R.C. Mullin, Inductive methods for balanced incomplete block designs, Ann. Math. Statist. 37 (1966) 1348-1354. MR 33 #5062; Z 144, p. 421. R.G. Stanton and R.C. Mullin, Uniqueness theorems in balanced incomplete block designs, J. Combinatorial Theory 7 (1969) 37-48. MR 39 #81; Z 172, p. 15. R.G. Stanton, see [H53], [Kl]. D. Steedley, Separable quasigroups, Aequat. Math. 11 (1974) 189-195. MR 53 #8310; Z 193.20058. S.K. Stein, Homogeneous quasigroups, Pacific J. Math. 14 (1964) 1091-1102. MR 30 #1206; Z 132, p. 265. J. Steiner, Combinatorische Aufgabe, J. reine angew. Math. 45 (1853) 181-182 (Gesammelte Werke, 11, 435-438). J. Steiner, Sur les dependances mutuelles des tangentes doubles des courbes du quatrieme degr6, C. R. Acad. Sci. Paris 37 (1853) 121-126. J. Steiner, Eigenschaften der Curven vierten Grades riicksichtlich ihrer Doppeltangenten. J. reine angew. Math. 49 (1855) 265-272 (Gesammelte Werke, 11. 603-612). F. Sterboul, see [D48.1]. J.J. Stiffler and I.F. Blake, An infinite class of generalized Room squares, Discrete Math. 12 (1975) 159-163 MR 51 #5343; Z 317.05018. J.J. Stiffler, see [B44]. T. Storer, see [J3], [J4]. T.H. Straley, Construction of Steiner quasigroups containing a specified number of subquasigroups of a given order, J. Combinatorial Theory (A) 13 (1972) 374-382. MR 46 #3672; Z 248.20092. T.H. Straley, see [L44], [L45], LL461. A. Suppa, see [F5]. J.D. Swift, Isomorph rejection in exhaustive search techniques, Proc. Symp. Appl. Math. 10 (1960) 195-200 (Amer. Math. SOC.,Providence, RI 1960) MR 22 #5145; Z 96, p. 5. J.D. Swift, Quasi-Steiner systems, Atti Accad. Naz. Lincei Rend. CI. Fis. Mat. natur. (8) 44 (1968) 40-44. MR 37 #6196; Z 159, p. 303.
344
[SS7]
[SbO]
J. Doyen, A. Rosa
J.D. Swift, A generalized Steiner problem, Rend. Mat. (6) 2 (1969) 563-569. MR 40 #7125; Z 322.05014. J.D. Swift, On (k, 1)-coverings and disjoint systems, Proc. Symp. Pure Math. 19 (Amer. Math. SOC..Providence, RI, 1971) 223-228. MR 47 #3212. J.D. Swift, see [H17], [H18]. [H19]. J.J. Sylvester, Elementary researches in the analysis of combinatorial aggregation, London, Edinburgh and Dublin Philos. Mag. and J. Sci. (3) 24 (1844) 285-296 (Collected Mathematical Papers. 1. 91-102). J.J. Sylvester, Note on the historical origin of the unsymmetrical six-valued function of six letters, London, Edinburgh and Dublin Philos. Mag. and J. Sci. 21 (1861) 369-377 (Collected Mathematical Papers, 11. 264-271). J.J. Sylvester. On a problem in tactic which serves to disclose the existence of a four-valued function of three sets of three letters each, London, Edinburgh and Dublin Philos. Mag. and J. Sci. (4) 21 (1861) 515-520 (Collected Mathematical Papers, 11, 272-276). J.J. Sylvester, Concluding paper on tactic, London, Edinburgh and Dublin Philos. Mag. and J. Sci. (4) 22 (1861) 45-54 (Collected Mathematical Papers, 11, 277-285). J.J. Sylvester, Remark on the tactic of nine elements, London, Edinburgh and Dublin Philos. Mag. and J. Sci. (4) 22 (1861) 144-147 (Collected Mathematical Papers 11, 286-289). J.J. Sylvester, Proposed problem, Educational Times (November 1 1875) 193. J.J. Sylvester, On the fifteen young ladies problem, Proc. London Math. Soc. 7 (1875-76) 235-236. J.J. Sylvester, Solution of a problem. Educational Times Reprints 33 ( I X X O ) 53 J.J. Sylvester, Note on a nine schoolgirls problem, Messenger Math. (2) 22 (1892-93) 159-160. Correction: 192. (Collected Mathematical Papers, IV, 732-733). L. Szamkolowicz, On the problem of existence of finite regular planes, Colloq. Math. 9 (1962) 245-250. MR 25 #5441; Z 106, p. 143. L. Szamkolowicz, Remarks on finite regular planes, Colloq. Math. 10 (1963) 31-37. MR 29 #1571; 2 118, p. 152. L. Szamkolowicz, Sur une classification des triplets de Steiner, Atti Accad. Naz. Lincei Rend. CI. Sci. Fis. Mat. Natur. (8) 36 (1964) 125-128. MR 31 #660; Z 137, p. 147. L. Szamkolowicz, Alcuni problemi della teoria dei sistemi di Steiner, Rend. Mat. e Appl. (5) 24 (1965) 348-359. MR 34 #7387; 2 146. p. 14. L. Szamkolowicz. Sulla generalizzazione del concetto delle algebra A:. Atti Accad. Naz. Lincei Rend. Cn. Ssi. Fis. Mat. Natur. (8) 38 (1965) 810-814. MR 33 #2753. L. Szamkolowicz, Remarks on algebras and regular finite planes, Colloq. Math. 17 (1967) 203-206. MR 36 #5255: Z 153, p. 338. L. Szamkolowicz. On Steiner manifolds, Colloq. Math. 20 (1969) 45-51. MR 39 #1584. K. Takeuchi, A table of difference sets generating balanced incomplete block designs, Rev. Inst. Internat. Statist. 30 (1962) 361-366. MR 28 #1712; Z 109, p. 122. P. Tannenbaum, Abelian inverse property neofields, Proc. Conf. Algebraic Aspects of Combinatorics (Toronto 1975). Congressus Numerantium XI11 (Utilitas Math., Winnipeg 1975) 279-281. MR 52 #5842; Z 318.12112. P. Tannenbaum, Abelian Steiner triple systems. Canad. J. Math. 28 (1976) 1251-1268. MR 54 #10468; 2 353.05012. P. Tannenbaum. Regular Steiner triple systems, Proc. 2nd Caribbean Conf. Combinatorics and Computing, U.W.I., Cave Hill, Barbados, 1977 (ed. R.C. Read and C.C. Cadogan) 2 15-218. L. Teirlinck, On the maximum number of disjoint Steiner triple systems, Discrete Math. 6 (1973) 299-300. MR 48 #5885; Z 266.05006. Id. Teirlinck, The existence of reverse Steiner triple systems, Discrete Math. 6 (1973) 301-302. MR 48 #8263; 2 266.05007. L. Teirlinck. On linear spaces on which every plane is either projective or affine, Geometriae Dedicata 4 (1975) 39-44. MR 52 #5441; Z 309.50014. L. Teirlinck, A simplification of the proof of the existence of reverse Steiner triple systems of order congruent to 1 modulo 24, Discrete Math. 13 (1975) 297-298. MR 51 #12562; 2 31 1.05011. I
Bibliography and survey of Steiner systems
345
L. Teirlinck, On the maximal number of disjoint triple systems, J. Geometry 6 (1975) 93-96. MR 51 #5333; Z 289.05016; Z 302.05011. [TI01 L. Teirlinck, Combinatorial Structures, Thesis, Vrije Universiteit Brussel, Departement voor Wiskunde, 1976, 106 pp. [Tll] L. Teirlinck, On making two Steiner triple systems disjoint, J. Combinatorial Theory (A) 23 (1977) 349-350. MR 56 #8394. [T12] L. Teirlinck, On Steiner spaces, J. Combinatorial Theory (A) 26 (1979) 103-1 14. [TI31 L. Teirlinck, Classification of small Steiner spaces (to appear). [TI41 L. Teirlinck, Planes and hyperplanes of 2-coverings, Bull. Soc. Math. Belg. 29 (1977) 73-81. rT14.11 L. Teirlinck, On projective and affine hyperplanes (to appear). L. Teirlinck, see [G4]. J.G. Thompson, see [M4]. J. Timm, see [H42]. J. Tits, Sur les systkmes de Steiner associts aux trois “grands” groupes de Mathieu, Rend. Mat. e Appl. (5) 23 (1964) 166-184. MR 32 #1262; Z 126, p. 263. J.A. Todd, On representations of the Mathieu groups as collineation groups, J. London Math. Soc. 34 (1959) 406-416. MR 22 #11045. J.A. Todd, A representation of the Mathieu group MZ4as a collineation group, Ann. Mat. Pura Appl. (4) 71 (1966) 199-238. MR 34 #2713; Z 144, p. 262. J.A. Todd, A representation of the Mathieu group MZ4as a collineation group, Rend. Mat. e Appl. (5) 25 (1966) 29-32. Z 156, p. 31. C.A. Treash, A method of obtaining Steiner triple systems, Ann. N. Y. Acad. Sci. 175 (1970) 383-384. MR 42 #97. C.A. Treash, The completion of finite incomplete Steiner triple systems with applications to loop theory, J. Combinatorial Theory (A) 10 (1971) 259-265. MR 43 #397; Z 217, p. 20. M.I. Trofimceva, see [AS]. L.E. Trotter Jr., see [F21]. J.T. Udding, see [H3], [H4]. G. Valette, see [D45]. R.M. Vancko, The spectrum of some classes of free universal algebras, Algebra Univ. 1 (1971) 46-43. M. Vandensavel, see [D43], [D46]. S.A. Vanstone, see [M9.1]. B.N. Varma, see [A10.1]. 0. Veblen, On magic squares, Messenger Math. 37 (1907-08) 116-1 18. 0. Veblen and W.H. Bussey, Finite projective geometries, Trans. Amer. Math. SOC.7 (1906) 241-259. J. Vild, see “191. H. Vogt, Analyse combinatoire et thkorie des dtterminants, Encycloptdie des Sciences mathkmatiques pures et appliqutes, tome I vol. 1, fasc. 1 (1904). Ternes ou triades, pp. 79-83. D. Volny, Defects of Steiner triple systems (to appear). H.L. de Vries, On property B and on Steiner systems, Math. Z. 153 (1977) 155-159. Z 332.05012. J. de Vries, Zur Theorie der Tripelsysteme, Rend. Circ. Mat. Palermo 8 (1894) 222-226. V. VuEkovit, Note on a theorem of W. Sierpiliski, Notre Dame J. Formal Logic 6 (1965) 180-182. MR 33 #3949; Z 201, p. 338. R. Walker, see [H19]. W.G. Warnock, Triple systems as ruled quadrics, Bull. Amer. Math. SOC.45 (1939) 476-480. Z 21, p. 151. T.M. Webb, Some constructions of sets of mutually almost disjoint Steiner triple systems, M.Sc. Thesis, Auburn University, June 1977. [W2.1] T.M. Webb, On the reconstruction problem of Steiner quadruple systems (to appear). [W3] E. de Weck, Sur les systbmes de quadruples, These prksentie a la Facultt des Sciences de I’Universitk de Fribourg, 1935. E. de Weck, see [B25].
[T9]
346
J. Doyen, A. Rosa
D.J.A. Welsh, Matroid Theory (Academic Press, London, 1976) Chapter 12: Block designs, pp. 193-216. H. Werner, A unique factorization theorem for Steiner triple systems, Technische Hochschule Darmstadt Preprint Nr. 19, June 1972. H. Werner, see [G5], [G6], [M22], [M26]. H.S. White. Triple-systems as transformations, and their paths among triads. Trans. Amer. Math. SOC.14 (1913) 6-13. H.S. White. The synthesis of triad systems A, in f elements, in particular for f = 31, Proc. Nat. Acad. Sci. U.S.A. 1 (1915) 4-6. H.S. White. The multitude of triad systems on 31 letters. Trans. Amer. Math. SOC.16 (1915) 13-19. H.S. White. Construction of a groupless and headless triad system on 31 elements, Bull. Amer. Math. SOC.40 (1934) 829-832. Z 10, p. 289. H.S. White, F.N. Cole and L.D. Cummings, Complete classification of the triad systems on fifteen elements, Memoirs Nat. Acad. Sci. U.S.A. 14, 2nd memoir (1919) 1-89. H.S. White, see [C21], [C351. R.M. Wilson, Cyclotomy and difference families in elementary abelian groups, J. Number Theory 4 (1972) 17-47. MR 46 #8860; Z 259.05011. R.M. Wilson. An existence theory for pairwise balanced designs. I.: Composition theorems and morphisms, J . Combinatorial Theory (A) 13 (1972) 220-245. MR 46 #3338; Z 263.050 14. R.M. Wilson, An existence theory for pairwise balanced designs, 11.: The structure of PBD-closed sets and the existence conjectures. J . Combinatorial Theory (A) 13 (1972) 246-273. MR 46 #3339; Z 263.05015. ~ 1 4 1R.M. Wilson. Nonisomorphic Steiner triple systems, Math. 2. 135 (1974) 303-313. MR 49 #4803; Z 264.05009. [WlS] R.M. Wilson, Some partitions of all triples into Steiner triple systems, Hypergraph Seminar (Ohio State University 1972), Lecture Notes in Math. 41 1 (Springer, Berlin, 1974) 267-277. MR 51 #5336; Z 311.05010. R.M. Wilson. An existence theory for painvise balanced designs, 111: Proof of the existence conjecture, J. Combinatorial Theory (A) 18 (1975) 7 1-79. Z 295.05002. R.M. Wilson. see [B4], [D47], [H30], [R4], [R5], [R6]. E. Witt. Die 5-fach transitiven Gruppen von Mathieu, Abh. Math. Sem. Univ. Hamburg 12 (1938) 256-264. Z 19, p. 251. E. Witt. Uber Steinersche Systeme, Abh. Mat. Sem. Univ. Hamburg 12 (1938) 265-275. Z 19. p. 251. M. Wojtas, On non-isomorphic balanced incomplete block designs B(4, 1, c ) , Colloq. Math. 35 (1976) 327-330. Wojtas. see [R18]. H.J. Woodall, see [Kl6]. W.S.B. Woolhouse, h i z e question 1733, Lady's and Gentleman's Diary (1844). W.S.B. Woolhouse, On the Rev. T.P. Kirkman's problem respecting certain triadic arrangements of fifteen symbols, London. Edinburgh and Dublin Philos. Mag. and J . Sci. (4) 22 (1861) 510-515. W.S.B. Woolhouse, On triadic combinations of 15 symbols, Lady's and Gentleman's Diary (1R62) 84-88 (reprinted in Assurance Magazine 10 (1862) 275-281). W.S.B. Woolhouse, On triadic combinations, Lady's and Gentleman's Diary ( 1863) 79-90. W.S.B. Woolhouse and S. Bills. Solution of a problem, Educational Times Reprints 8 (1867) 76-83. C.R. Wylie. see [H33]. C.C. Yalavigi, Kirkman's school girls problem, Math. Education 4 (1970) 15-18. MR 42 #7531. C.C. Yalavigi, Direct construction of Kirkman's designs for k 3. Math. Education, Sect. A 12 (1978) 28-30. Z 388.05001. F. Yates, see [F9]. 2
Bibliography and survey of Sreiner systems
347
H.P. Young, Affine triple systems, Finite Geometric Structures and their Applications (C.I.M.E., I1 Ciclo Bressanone 1972) (Edizjoni Cremonese, Roma, 1973) 265-282. MR 50 #142a; Z 297.05031. H.P. Young, Affine triple systems and matroid designs, Math. Z. 132 (1973) 343-359. MR 50 #142b; Z 263.05019. H.P. Young and J. Edmonds, Matroid designs, J. Res. Nat. Bur. Standards 77B (1973) 15-44. MR 52 #2929. G.V. Zaicev, V.A. Zinoviev and N.V. Semakov, Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes, Proc. 2nd Internat. Sympos. Information Theory, Tsahkadsor, Armenia, USSR, 1971 (Akademiai Kiado, Budapest, 1973) 257-263. H. Zeitler, Konstruktion spezieller Steiner-Tripel-Systeme, Math. -Phys. Semesterber. 21 (1974) 206-233. MR 52 #132. L. Zhu, A construction for orthogonal Steiner triple systems, Ars Combinatoria (to appear). V.A. Zinoviev, see [ZO]. A. Zirakzadeh, A model for the finite projective spaces with three points on every line, Amer. Math. Monthly 76 (1969) 774-778. MR 40 #3419; Z 181, p. 233. K. Zulauf, Uber Tripelsysteme von 13 Elementen, Dissertation Giessen, Wintersche Buchdruckerei. Darmstadt 1897.
2. A survey of “small” Steiner systems Given three integers t, k, u such that 2 s t < k < u, a Steiner system S(t, k, u ) is a u-set S together with a family B of k-subsets of S (called blocks) such that any t-subset of S is contained in exactly one block. The existence of an S(t, k, u) implies that
is an integer for every i = 0, 1,. . . , t - 1; the parameters t, k and u are called admissible if they satisfy this necessary condition for existence. Below, we extend the listing of all sets of admissible parameters for S(t, k, u ) to those with u ~ 2 8 Information . on N(r, k, u ) , the number of pairwise nonisomorphic systems S ( t , k, u ) , is also included. The numbers in square brackets refer to the titles in the bibliography.
t‘
7 8 9 10 11 12 13 14
S(t, k, v )
Existence
Nt, k, u)
Yes Yes Yes Yes Yes Yes Yes Yes Yes
1 1 1 1 1 1
2
References
[B91 [B9][W18l[L5 I] [B9][ W 18][L5 11 [B9KW18][LS 11 [D 18][B67][C19]
1
4
[M201
1. Doyen, A. Rosa
348
u
1s
16
17
18
19
20
21
22
23
24
25
26
27
S(f,k . v ) S(2,3. I S ) S(4.S. IS) S(2.4. 16) S(2.6, 16) S(3.4, 16) S(S. 6. 16) S(3.5, 17) Sl4,5. 17) S(6.7, 17) S ( 4 6 , 18) S(5.6, 18) S(7,X. 18) S(2-3.19) S(6,7, 19) S(8.9. 19) S(3,4,20) S(7.8.20) S(9. 10.20) S(2,3.21) S(2.5,21) S(2.6.21) S(4, 5.21 J S(X,9,2I I S(3,4,22) S(3, ti. 22) SI3.7.22) 35.6.22) S ( 9 . 10.22) S(4. S. 23) S(4,7.23) S(6, 7.23) S(10. 11.23) S(S.6.24) S(S. 8.24) S(7,8.24) S(11.12.24) S(2.3.25) S(2.4.25) S(2.5, '5) S(6.7.25) S(8.9.25) S(3.4.26) S(3.5. 26) S(3.6.26) S(7. X. 26) S(9.10.261 S(2.3.27) S ( 4 . S .27) S(4,6.27) S ( 8 . 9 , 271 S(10, 11,27)
Existence Yes No Yes No Yes No Yes
N(r, k , c )
80 0 1
References [C2 l][WlQ][F7][H17] w2qi [WW
0 '? 2 3 130 1 0 1
)
No NO
0 0
? NO
0
Yes
? 2 284407
No Yes 1
No Yes Yes No
0 ? 3 2 160980
1 0
0 )
Yes Yes No
?221 1
0
)
.? Yes Yes
>
1
9
?
Yes Yes
?22 1
? ?
Yes Yes Yes ? '?
Yes Yes Yes >
'?
Yes )
Yes '? *?
? 2 10'4 .? 2 4 1
~ 4 1 [R17]. A.E. Brouwer ~ 3 1
Bibliography and survey of Steiner systems
V
S(t, k , li)
Existence
N(t, k, u)
References
349
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