On Ramsey-type theorems

In [21], Frank Plumpton Ramsey proved what has turned out to be a remarkable and important theorem which is now known as Ramsey's theorem. This result is a generalization of the pigeonhole principle and can now be seen as part of a family of theorems of the same flavour. These Ramsey-type theorems all have the common feature that they assert, in some precise combinatorial way, that if we deal with large enough sets of numhers, there will be some well behaved fragment in the set. In Harrington's words, Ramsey-type theorems assert that complete chaos is impossible. Ramsey-type theorems have turned out to be very important in a number of branches of mathematics. In this paper we shall survey a number of basic Ramsey-type theorems, and we will then look at a selection of applications of Ramsey-type theorems and Ramsey-type ideas. In the applications we will concentrate on graph theory, logic and complexity theory. Proofs will mostly not be given in detail, but it is hoped that the reader will gain some appreciation of the usefulness and importance of the beautiful area of asymptotic combinatorics.