In this chapter model theory will play a more prominent role. In the noughth section we shall lay the model-theoretic foundations for the subsequent chapters and introduce the notions of “foreign” and “internal” due to Hrushovski. This allows us to generalize the concept of “small” (e.g. finite) and “large” (e.g. generic) sets: a definable subset is “small” if the generic type is foreign to it, and “large” if the generic type is internal or analysable in it. It will turn out in section 1 that these notions are particularly well-behaved in the context of groups: if the generic type of a group is not foreign to some set X, then there is a definable X-internal quotient of the group. This means that quite often it is sufficient to consider only definable subsets which define groups. Furthermore there is some form of compactness: if a group is internal in a class of definable sets, then it is internal in a subset of them, of bounded cardinality.

In section 2, we shall consider various components of a stable group. In particular we shall define the Φ-component, which enjoys very strong connectivity properties. It is the intersection of all relatively definable subgroups not only of finite, but of “small” index (defined in terms of analysability of the whole group using the quotient), and it generalizes both the characteristic normal subgroup of monomial U-rank given by Berline and Lascar, and Hrushovski's p-connected component (where p is a regular type in which the group is internal).

Abstract

We survey some recent classification theorems for elation generalized quadrangles of order (q2, q), q even, with particular emphasis on those involving subquadrangles of order q.

Abstract

If x is a regular point of a generalized quadrangle S = (P, B, I) of order (s, t), s ≠ 1, then x defines a dual net with t + 1 points on any line and s lines through every point. If s ≠ t, s > 1, t > 1, then S is isomorphic to a T3(O) of Tits if and only if S has a coregular point x such that for each line L incident with x the corresponding dual net satisfies the Axiom of Veblen. As a corollary we obtain some elegant characterizations of the classical generalized quadrangles Q(5, s). Further we consider the translation generalized quadrangles S(p) of order (s, s2), s ≠ 1, with base point p for which the dual net defined by L, with p I L, satisfies the Axiom of Veblen. Next there is a section on Property (G) and the Axiom of Veblen, and a section on flock generalized quadrangles and the Axiom of Veblen. This last section contains a characterization of the TGQ of Kantor in terms of the Axiom of Veblen. Finally, we prove that the dual net defined by a regular point of S, where the order of S is (s, t) with s ≠ t and s ≠ 1 ≠ t, satisfies the Axiom of Veblen if and only if S admits a certain set of proper subquadrangles.

Abstract

In the last few years there has been rapid progress in the theory of difference sets. This is a survey of these fascinating new developments.

Abstract

The ternary codes associated with the five known biplanes of order 9 were examined using the computer language Magma. The computations showed that each biplane is the only one to be found among the weight-11 vectors of its ternary code, and that none of the biplanes can be extended to a 3-(57,12,2) design. The residual designs of the biplanes, and designs associated with {12; 3}-arcs were also examined.

Abstract

It is shown that the function field of the algebraic envelope of the cyclic (q — √q +1) –arc in PG(2, q), q square, is the Hermitian function field.

Abstract

We consider the following problem: given a partial geometry with v points and k points on a line, can one add to the line set a set of k-subsets of points such that the extended family of k-subsets is a 2-(v, k;, 1) design (or a Steiner system S(2, k, v)). We give some necessary conditions for such embeddings and several examples. One of these is an embedding of the partial geometry PQ+(7,2) into a 2-(120,8,1) design.

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.