Ramsey’s theorem for $n$-parameter sets

Abstract:

Classes of objects called $n$-parameter sets are defined. A Ramsey theorem is proved to the effect that any partitioning into $r$ classes of the $k$-parameter subsets of any sufficiently large $n$-parameter set must result in some $l$-parameter subset with all its $k$-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.