Multiply partition regular matrices

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 A~x 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 A~x + B~y = ~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 = ( A cB ) is kernel partition regular. (That is, whenever N is finitely coloured, there exists monochromatic ~x such thatM~x = ~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. Email addresses: [email protected] (Dennis Davenport), [email protected] (Neil Hindman), [email protected] (Imre Leader), [email protected] (Dona Strauss) This author acknowledges support received from the National Science Foundation (USA) under grant DMS-1160566. We also prove extensions to the case of several matrices.