Partition regularity of ( M, P, C)-systems

Abstract

By an ( M, P, C)-system we mean a choice of an ( m, p, c)-set for each ( m, p, c) ϵ N 3 together with all finite sums choosing at most one from each ( m, p, c)-set. We show here that ( M, P, C)-systems are partition regular. That is, if an ( M, P, C)-system is partitioned into finitely many cells, one of these cells contains an ( M, P, C)-system. We also prove a multidimensional version similar to the Milliken-Taylor version of the Finite Sum Theorem.