Ramsey's theorem for a class of categories

Abstract

In this paper we present a Ramsey theorem for certain categories which is sufficiently general to include as special cases the finite vector space analog to Ramsey’s theorem (conjectured by Gian-Carlo Rota), the Ramsey theorem for n-parameter sets [21, as well as Ramsey’s theorem itself [4, 61. The Ramsey theorem for finite affine spaces is obtained here simultaneously with that for vector spaces. That these two are equivalent was already known [5, II, and the arguments previously used to show that the affine theorem implies the projective theorem are also special cases of the results of this paper.