Nonstandard combinatorics

Ramsey type theorems are theorems of the form: if certain sets are partitioned at least one of the parts has some particular property. In its finite form, Ramsey's theory will ask how big the partitioned set should be to assure this fact. Proofs of such theorems usually require a process of multiple choice, so that this apparently pure combinatoric field is rich in proofs that use ideal guides in making the choices. Typically they may be ultrafilters or points in the compactification of the given set. It is, therefore, not surprising that nonstandard elements are much more natural guides in some of the proofs and in the general abstract treatment.In Section 1 we start off with some very natural examples of Ramsey type exercises that illustrate our idea. In Section 2 we give a nonstandard proof of the infinite Ramsey theorem. Section 3 tries to do the same for Hindman's theorem, and points out, where nonstandard analysis must use some hard standard facts to make the proof go through.In Section 4 we describe a general theory of Ramsey Properties, identifying a Ramsey Property with its nonstandard kernel in the enlargement.In Section 5 nonstandard analysis is used again to deduce the finite Ramsey theorems from their infinite counterpart. More generally, a compactness theorem is proved to work for all theorems of this type.