We first give two different formulations of a theorem known as P. Hall's marriage theorem. We give a constructive proof and an enumerative one. If A is a subset of the vertices of a graph, then denote by г(A) the set ∪a∈Aг(a). Consider a bipartite graph G with vertex set X ∪ Y (every edge has one endpoint in X and one in Y). A matching in G is a subset E1 of the edge set such that no vertex is incident with more than one edge in E1. A complete matching from X to Y is a matching such that every vertex in X is incident with an edge in E1. If the vertices of X and Y are thought of as boys and girls, respectively, or vice versa, and an edge is present when the persons corresponding to its ends have amicable feelings towards one another, then a complete matching represents a possible assignment of marriage partners to the persons in X.

Theorem 5.1.A necessary and sufficient condition for there to be a complete matching from X to Y in G is that |г(A) > |A| for every A ⊆ X.

Proof: (i) It is obvious that the condition is necessary.

(ii) Assume that |г(A)| ≥ |A| for every A ⊆ X. Let |X| = n, m < n, and suppose we have a matching M with m edges. We shall show that a larger matching exists.

Given two k-subsets A, B of an n-set, n ≥ 2k, there are k + 1 possible relations between them: they may be equal, they may intersect in k − 1 elements, they may intersect in k − 2 elements,…, or they may be disjoint.

Given two words (k-tuples) a, b ∈ Ak, where A is an ‘alphabet’ of size at least 2, there are k + 1 possible relations between them: they may be equal, they may agree in k − 1 coordinates, they may agree in k − 2 coordinates,…, or they may disagree in all coordinates.

These instances of a set together with a list of mutually exclusive and exhaustive binary relations are examples of association schemes, which we define shortly. Association schemes provide one of the foundations of combinatorics and so we include this chapter even though it will be difficult reading. They have been implicit in many of the previous chapters; we have explicitly discussed 2-class association schemes, as they are equivalent to the strongly regular graphs discussed in Chapter 21. This chapter elaborates on some of the material of Chapter 21 but has different goals.

Association schemes arose first in the statistical theory of design of experiments, but the work of Ph. Delsarte (1973) has shown how they serve to unify many aspects of our subject. In particular, certain results of coding theory and the theory of t-designs—which were originally discovered independently—are now seen to be ‘formally dual’ aspects of the same ideas in association schemes.

Many beautiful and elegant results assert that if we partition a sufficiently large structure into k parts, then at least one of the parts contains a substructure of a given size. For example, Schur (1916) proved that if the natural numbers are partitioned into finitely many classes, then x + y = z is solvable in one class, and van der Waerden (1927) proved that one class of such a partition contains arbitrarily long arithmetic progressions. The quintessential partition theorem is the classical theorem of Ramsey (1930) which concerns very simple structures indeed: if for some r ∈ ℕ the set ℕ(r) of all r-subsets of ℕ is divided into finitely many classes then ℕ has an infinite subset all of whose r-subsets belong to the same class. All these statements have analogues for finite sets; these analogues tend to be more informative and are of great interest in finite combinatorics. The theory dealing with theorems in this vein has become known as Ramsey theory.

By now there is an immense literature on Ramsey theory; the popularity of the field owes a great deal to Paul Erdős, who proved many of the major results and who was the first to recognize the importance of partition theorems.

In this brief chapter we restrict our attention to Ramsey theorems concerning graphs whose proofs are based on the use of random graphs, so our treatment of the subject is far from encyclopaedic.