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.

The aim of this chapter is to present the definitions, formulae and results of probability theory we shall need in the main body of the book. Although we assume that the reader has had only a rather limited experience with probability theory and, if somewhat vaguely, we do define almost everything, this chapter is not intended to be a systematic introduction to probability theory. The main purpose is to identify the facts we shall rely on, so only the most important—and perhaps not too easily accessible—results will be proved. Since the book is primarily for mathematicians interested in graph theory, combinatorics and computing, some of the results will not be presented in full generality. It is inevitable that for the reader who is familiar with probability theory this introduction contains too many basic definitions and familiar facts, while the reader who has not studied probability before will find the chapter rather difficult.

There are many excellent introductions to probability theory: Feller (1966), Breiman (1968), K. L. Chung (1974) and H. Bauer (1981), to name only four. The interested reader is urged to consult one of these texts for a thorough introduction to the subject.

Notation and Basic Facts

A probability space is a triple (Ω, ∑, P), where Ω is a set, ∑ is a σ-field of subsets of Ω, P is a non-negative measure on ∑ and P(Ω) = 1.

The period since the publication of the first edition of this book has seen the theory of random graphs go from strength to strength. Indeed, its appearance happened to coincide with a watershed in the subject; the emergence in the subsequent few years of significant new ideas and tools, perhaps most notably concentration methods, has had a major impact. It could be argued that the subject is now qualitatively different, insofar as results that would previously have been inaccessible are now regarded as routine. Several longstanding issues have been resolved, including the value of the chromatic number of a random graph Gn,p, the existence of Hamilton cycles in random cubic graphs, and precise bounds on certain Ramsey numbers. It remains the case, though, that most of the material in the first edition of the book is vital for gaining an insight into the theory of random graphs.

It would be impossible, in a single volume, to prove all the substantial new results that we would wish to, so we have chosen instead to give brief descriptions and to sketch a number of proofs.