Partition regular inequalities (Q1266371)

A generalization of partition regular matrices is discussed. Given a rational \(m \times n\) matrix \(A\) which is partition regular, i.e., for every finite colouring of the natural numbers there is a monochromatic vector \(x\) with \(Ax = 0\), and given an additional constraint \(\sum b_i x_i > 0\), under what conditions does there exist a monochromatic vector \(x\) satisfying both \(Ax=0\) and \(\sum b_i x_i > 0\) for every finite coloring of the natural numbers? The authors generalize an earlier result by Rado who derived a sufficient condition for the above-mentioned property. In this paper a necessary and sufficient condition is derived for the more general case that \(d\) constraints (\(d \geq 1\)) of the type \(\sum b_i x_i > 0\) are added to the system \(Ax=0\). Furthermore the special case that \(A\) is empty is discussed, answering the question which matrices \(B\) satisfy that, for every finite colouring of the natural numbers, there exists a monochromatic vector \(x\) with \(Bx > 0\).