Ramsey’s theorem for 𝑛-parameter sets

Abstract

Classes of objects called n n -parameter sets are defined. A Ramsey theorem is proved to the effect that any partitioning into r r classes of the k k -parameter subsets of any sufficiently large n n -parameter set must result in some l l -parameter subset with all its k k -parameter subsets in one class. Among the immediate corollaries are the lower dimensional cases of a Ramsey theorem for finite vector spaces (a conjecture of Rota), the theorem of van der Waerden on arithmetic progressions, a set theoretic generalization of a theorem of Schur, and Ramsey’s Theorem itself.