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? enerally, 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 c B )  is kernel partition regular. (That is, whenever N is finitely coloured, there exists monochromatic x → such that M 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 − c I )  is kernel partition regular, where I is the identity matrix of the appropriate size. o prove extensions to the case of several matrices.