Universally Image Partition Regularity

Abstract

Many of the classical results of Ramsey Theory, for example Schur's Theorem, van der Waerden's Theorem, Finite Sums Theorem, are naturally stated in terms of image partition regularity of matrices. Many characterizations are known of image partition regularity over ${\Bbb N}$ and other subsemigroups of $({\Bbb R},+)$. In this paper we introduce a new notion which we call universally image partition regular matrices, which are in fact image partition regular over all semigroups and everywhere. We also prove that such matrices exist in abundance.