This volume contains articles based on talks given at the First Pythagorean Conference on Geometry, Combinatorial Designs and Related Structures, held on the island of Spetses in Greece from 1 to 7 June 1996.

There were 80 invited participants and 48 talks, including hour-long expositions by the keynote speakers: Peter Cameron, John Conway, Jean Doyen, Dieter Jungnickel, Curt Lindner, Rudi Mathon, Ernie Shult, Jef Thas.

It was a conference which will live long in the memory. Apart from the weather and the seaside setting, several other events contributed to an outstanding week. Jean Doyen opened the conference by giving a historical talk on Pythagoras. At a meeting of the Institute of Combinatorics and its Applications, Jef Thas was awarded the 1994 Euler Medal for a distinguished lifetime career. The excursion around the Peloponnesian peninsula included the theatre at Epidaurus and the hilltop ruins of Mycenae.

The editors, who were also the conference organisers, would like to thank the Institute of Combinatorics and its Applications, the Greek Ministry of Civilization, and the Greek Ministry of Education for their financial support. We would also like to thank the Greek Tourist Organization for providing a beautiful illustrated book of Greece for all the participants. Special thanks are due to Simos Magliveras for working out many organizational and tactical problems, and also to Sakis Simopoulos for considerable help in further organizational matters.

Abstract

Some simple 7-designs with small parameters are constructed with the aid of a computer. The smallest parameter set found is 7-(24, 8, 4). An automorphism group is prescribed for finding the designs and used for determining the isomorphism types. Further designs are derived from these designs by known construction processes.

Abstract

Studying the geometry of a group G leads us to questions about its maximal subgroups and primitive permutation representations (the G-invariant relations and similar structures, the base size, recognition problems, and so on). Taking the point of view that finite projective geometry is the geometry of the groups PGL(n, q), Aschbacher's theorem gives us eight natural families of geometric objects, with greater or smaller degrees of familiarity. This paper presents some speculations on how the subject could develop from this point of view.

Abstract

The counting of the number of Intercalates, 2 × 2 subsquares, possible in a latin square of side n is in general a hard problem. N2–Free latin squares, those for which there are no intercalates, are known to exist for n ≠ 1,2,4. N2–complete latin squares, those which have the property that they have the maximum number of N2's possible, must be isotopic to and thus of side 2k. The maximum for n ≠ 2k is in general unknown. We propose an intermediate possibility, that of N2–ubiquitous. A latin square is N2 ubiquitous if and only if every cell aij is contained in some 2 × 2 subsquare. We show these exist for n ≠ 1,3,5,7. It is also determined for which n, C–ubiquitous latin squares exist for every partial latin square, C with four cells. We also enumerate the number of times each 4-cell configuration can appear in a latin square and show that this number depends only on n and the number of intercalates.

Abstract

Given an n-set X, we denote the cardinality of a maximum size anti-Pasch (Pasch free) set of triples of X by f(n). In this paper we provide lower and upper bounds for f(n) and consequently we disprove a conjecture posed by Khosrovshahi at the fifteenth British Combinatorial Conference (BCC15).

Abstract

We show that there are no regular parallelisms of PG(3, 3) but that there are many parallelisms consisting entirely of subregular spreads of index one.

Abstract

The Veronese correspondence maps the set of all plane conics which are tangent to the sides of a given triangle in PG(2,q), q odd, to a (2q2 – q + 2)–cap in PG(5, q) obtained as the complete intersection of three quadratic cones. This cap can also be represented as the union of two quadric Veroneseans sharing three conics pairwise meeting at one point. Some information about the (setwise) stabilizer of this cap in PGL(6, q) is also given.

Abstract

A Rosa triple system is a triple system of order congruent to 2 modulo 3 whose chromatic index is minimum, having the largest possible number of maximum parallel classes in such a block colouring. The existence of Rosa triple systems is settled completely. Together with known results on Kirkman, Hanani, and almost resolvable twofold triple systems, the existence of Rosa triple systems is used to settle completely the existence of triple systems with minimum chromatic index.

Abstract

A double-five of planes is a set ψ of 35 points in PG(5, 2) which admits two distinct decompositions ψ = α1 ∪ α2 ∪ α3 ∪ α4 ∪ α5 = β1 ∪ β2 ∪ β3 ∪ β4 ∪ β5 into a set of five mutually skew planes such that αr ∩ βr is a line, for each r, while αr ∩ βs is a point, for r ≠ s. In a recent paper, [Sh96], a construction of a double-five was given, starting out from a (suitably coloured) icosahedron, and some of its main properties were described. The present paper deals first of all with some further properties of double-fives. In particular the existence of an invariant symplectic form is demonstrated and some related duality properties are described.

Secondly the relationship of double-fives to partial spreads of planes in PG(5, 2) is considered. The α-planes, or equally the β-planes, of double-fives provide the only examples of maximal partial spreads. It is shown that one of the planes of a non-maximal partial spread of five planes is always privileged, and this fact is seen to give rise to a nice geometric construction of an overlarge set of nine 3-(8, 4, 1) designs having automorphism group ΓL2(8).

Abstract

This is a survey of all known rank three geometries belonging to a string diagram of type (c*, c)-geometry. There are three types of objects: points, lines, and blocks subject to axioms imposed by the diagram. There are several other formulations described here which are more convenient for presenting certain of the examples. All examples fall into these six classes:

1. Simplicial type, which can easily be characterized.

2. Fischer spaces with no affine planes.

3. Orthogonal types, whose points and lines are exterior points and tangent lines of certain low-dimensional quadrics.

4. Hall type, determined by alternating multilinear forms over the field of two elements.

5. Affine type, whose points are vectors in some d-dimensional space over the integers mod 2. Here, blocks are not subspaces.

6. A few special examples determined by coherent pairs: the construction of odd type of Cameron and Fisher, and two examples of Blokhuis and Brouwer.

Abstract

New algorithms are presented for finding spreads and packings of sets with applications to combinatorial designs and finite geometries. An efficient deterministic method for spread enumeration is used to settle several existence problems for t-designs and partial geometries. Randomized algorithms based on tabu search are employed to construct new Steiner 5-designs and large sets of combinatorial designs. In particular, partitions are found of the 4-subsets of a 16-set into 91 disjoint affine planes of order 4.

Abstract

Buekenhout has given a construction of unitals in PG(2, q2) using the André representation of PG(2, q2) in the space PG(4,q). Metz has shown that this construction produces hermitian and non-hermitian unitals. In this note, we give a geometric criterion in PG(4, q) to decide whether the unital in PG(2, q2) is hermitian or not.

The 1997 issue of the British Combinatorial Bulletin contains a short history, written by Norman Biggs, of the early years of the British Combinatorial Conference. The first one was held at Oxford in 1969. The sixth conference, held at Royal Holloway College in 1977, was the first at which a volume containing the invited talks was published in time to be available to participants at the conference. Peter Cameron was the pioneering editor of that volume. Such a volume has been produced for every conference thereafter.

The 1977 conference was also the first one that I attended. There I joined the British Combinatorial Committee, which was formally set up at that meeting although it had effectively existed for some years—the previous conferences didn't just organize themselves. As often happens, I found that being on the committee considerably widened my knowledge of the subject. I left the committee in 1981, but have never lost touch with combinatorial activity in Britain.

I was delighted when I was asked to edit the present volume. In spite of the work involved, I am still delighted. I have had a preview of nine magnificent papers, and come to know their subject matter much better than I would otherwise have done.

At the centre of this volume is a long paper by Bruce Reed about the tree width of graphs. This is a new measure of connectivity. It is intimately linked to the concept of a minor of a graph, which is obtained by erasing an edge or coalescing two vertices joined by an edge, or by a sequence of such operations.

Summary The group M12 has no transitive extension, but the object of the title is the next best thing: a set of permutations which is an extension of M12. We give an elementary construction, based on a moving-counter puzzle on the projective plane of order 3, and provide easy proofs of some of its properties.

Introduction

Long ago I was intrigued by the fact that M12, É. Mathieu's celebrated quintuply transitive group on 12 letters, shares some structure with L3(3), which acts doubly transitively on the 13 points of the projective plane PG(2,3), of which it is the automorphism group.

To be more precise, the point-stabilizer in L3(3) is a group of structure 32: 2S4 that permutes the 12 remaining points imprimitively in four blocks of 3, and there is an isomorphic subgroup of M12 that permutes the 12 letters in precisely the same fashion. Again, the line-stabilizer in L3(3) is a group of this same structure that permutes the 9 points not on that line in a doubly transitive manner, while the stabilizer of a triple in M12 is an isomorphic group that permutes the 9 letters not in that triple in just the same manner.

In the heady days when new simple groups were being discovered right and left, this common structure inevitably suggested that there should be a new group that contained both M12 and L3(3), various copies of which would intersect in the subgroups mentioned above.