Multiply partition regular matrices

Abstract

Let $A$ be a finite matrix with rational entries. We say that $A$ is {\it doubly image partition regular\/} if whenever the set ${\mathbb N}$ of positive integers is finitely coloured, there exists $\vec x$ such that the entries of $A\vec x$ are all the same colour (or {\it monochromatic\/}) and also, the entries of $\vec x$ 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 {\it doubly kernel partition regular\/} if whenever ${\mathbb N}$ is finitely coloured, there exist vectors $\vec x$ and $\vec y$, each monochromatic, such that $A \vec x + B \vec y = 0$. 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 $M=(\begin{array}{ccccc}A&cB\end{array})$ is kernel partition regular. (That is, whenever ${\mathbb N}$ is finitely coloured, there exists monochromatic $\vec x$ such that $M \vec x=\vec 0$.) 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 $(\begin{array}{lr}A&cI\end{array})$ is kernel partition regular, where $I$ is the identity matrix of the appropriate size. We also prove extensions to the case of several matrices.