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.