R. M. Wilson

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ0, μ1, ..., μs are distinct residues modp (p is a prime) such thatk ≡ μ0 (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μi (modp) for somei, 1 ≦i≦s, then |ℱ|≦(sn).As a consequence we show that ifRn is covered bym sets withm<(1+o(1)) (1.2)n then there is one set within which all the distances are realised.It is left open whether the same conclusion holds for compositep.

Quasi-random graphs

We introduce a large equivalence class of graph properties, all of which are shared by so-called random graphs. Unlike random graphs, however, it is often relatively easy to verify that a particular family of graphs possesses some property in this class.

Cyclotomy and difference families in elementary abelian groups

By a (v, k, λ)-difference family in an additive abelian group G of order v, we mean a family (Bi ∣ i ∈ I) of subsets of G, each of cardinality k, and such that among the differences (a − b ∣ a, b ∈ Bi; a ≠ b; i ∈ I) each nonzero g ∈ G occurs λ times. The existence of such a difference family implies the existence of a (v, k, λ)-BIBD with G as a regular group of automorphisms. The multiplicative structure of finite fields is used here to construct difference families in their additive groups. It is proven that if q is a prime power and q > {12k(k − 1)}k(k−1), then there exists a (q, k, λ)-difference family in the elementary abelian group of order q iff λ(q − 1) ≡ 0 (mod k(k − 1)). Some extensions of known constructions and partially balanced designs are also discussed.

Constructions and Uses of Pairwise Balanced Designs

A pairwise balanced design (PBD) of index unity is a pair (X,A) where X is a set (of points) and A a class of subsets A of X (called blocks) such that any pair of distinct points of X is contained in exactly one of the blocks of A (and we may also require |A| ≥ 2 for each A ∈ A). Such systems are also known as linear spaces. PBD’s where all blocks have the same size |A| = k are known as balanced incomplete block designs (BIBD’s) of index λ = 1, as 2 - (v,k,1) designs, and as Steiner systems S(2,k,v). The more general concept, where multiple block sizes are allowed, was introduced by BOSE, Shrikhande & Parker [4] and H. Hanani [9], and played important roles in their respective work on orthogonal Latin squares and Bibd’s.

An existence theory for pairwise balanced designs, III: Proof of the existence conjectures

Abstract Given positive integers k and λ, balanced incomplete block designs on v points with block size k and index λ exist for all sufficiently large integers v satisfying the congruences λ ( v − 1) ≡ 0 (mod k − 1) and λv ( v − 1) ≡ 0 (mod k ( k − 1)). Analogous results hold for pairwise balanced designs where the block sizes come from a given set K of positive integers. We also observe that the number of nonisomorphic designs on v points with given block size k > 2 and index λ tends to infinity as v increases (subject to the above congruences).

An existence theory for pairwise balanced designs I. Composition theorems and morphisms

One of the most central problems of modern combinatorial theory is the determination of those parameter triples (v, k, X) for which there exist (0, k, h)-BIBD’s (balanced incomplete block designs). The methods which have been put forth to construct BIBD’s divide roughly into two classes: direct constructions where a BIBD is obtained from an algebraic structure (often a design is constructed from its automorphism group), and recursive or composition methods where a BIBD is built up by purely combinatorial means from another design or an assortment of “smaller” designs (see [6]). It is the second class with which we deal here. One of the first instances of a composition theorem was presented by E. H. Moore [l l] in 1893 in connection with Steiner triple systems, i.e., (0, 3, I)-BIBD’s. Several methods were expounded by Bose and Shrikhande [l] and composition techniques were instrumental in their remarkable work (with E. T. Parker) on orthogonal Latin squares [2, 31. Significant contributions have been made by H. Hanani [7, 8, lo] in his work on BIBD’s with k = 3,4, 5 and composition methods are used by Ray-Chaudhuri and Wilson [12] in connection with Kirkman designs. An attempt is made here to unify some of these various constructions and to present them in a more general, common setting. The general theorems presented here will be illustrated by examples and will be applied in the second part of this article, “An Existence Theory for Pairwise Balanced Designs, II. The Structure of PBD-Closed Sets and the Existence Conjectures” [15], to a conjecture on the existence of BIBD’s [6, p. 2381. For simplicity of exposition, we consider only designs with X = 1. We

On the minimum distance of cyclic codes

The main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds (i.e., BCH bound, HT bound, Roos bound). This bound is related to a second method for bounding the minimum distance of a cyclic code, which we call shifting. This method can be even stronger than the first one. For all binary cyclic codes of length (with two exceptions), we show that our methods yield the true minimum distance. The two exceptions at the end of our list are a code and its even-weight subcode. We treat several examples of cyclic codes of length \geq 63 .

The exact bound in the Erdös-Ko-Rado theorem

AbstractThis paper contains a proof of the following result: ifn≧(t+1)(k−t−1), then any family ofk-subsets of ann-set with the property that any two of the subsets meet in at leastt points contains at most

An existence theory for pairwise balanced designs II. The structure of PBD-closed sets and the existence conjectures☆

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