Multiply partition regular matrices

Abstract

Let A be a finite matrix with rational entries. We say that A is doubly image partition regular if whenever the set N of positive integers is finitely coloured, there exists x→ such that the entries of Ax→ are all the same colour (or monochromatic) and also, the entries of x→ are monochromatic. Which matrices are doubly image partition regular?More generally, we say that a pair of matrices (A,B), where A and B have the same number of rows, is doubly kernel partition regular if whenever N is finitely coloured, there exist vectors x→ and y→, each monochromatic, such that Ax→+By→=0→. (So the case above is the case when B is the negative of the identity matrix.) There is an obvious sufficient condition for the pair (A,B) to be doubly kernel partition regular, namely that there exists a positive rational c such that the matrix M=(AcB) is kernel partition regular. (That is, whenever N is finitely coloured, there exists monochromatic x→ such that Mx→=0→.) Our aim in this paper is to show that this sufficient condition is also necessary. As a consequence we have that a matrix A is doubly image partition regular if and only if there is a positive rational c such that the matrix (A−cI) is kernel partition regular, where I is the identity matrix of the appropriate size.We also prove extensions to the case of several matrices.