Ramsey's Theorem for a Class of Categories

Abstract

Ramsey's Theorem states that for a sufficiently large set S, and for any splitting of the k-element subsets of S into r classes, there is a subset T [unk] S, [unk]T[unk] = l, such that all k-element subsets of T are in the same class. This paper establishes a theorem for certain categories that generalizes Ramsey's Theorem. In particular, it is strong enough to establish G-C. Rota's conjecture that the vector space analogue to Ramsey's Theorem is true. It also implies the Ramsey theorem for n-parameter sets, which has as corollaries, among others, the theorem of van der Waerden on arithmetic progressions and several results of R. Rado on regularity in systems of linear equations.