Partition Regular Inequalities

Anm × nrational matrixAis said to be partition regular if for every finite colouring of N there is a monochromatic vectorxwithAx = 0. Given a partition regular matrixA, and a linear inequality such as ∑ bixi > 0, when is it true that for every finite colouring of N there is a monochromatic vectorxsatisfyingAx = 0 and the additional constraint ∑ bixi > 0? This is certainly the case if we can guarantee that ∑ bixi(or some fixed positive multiple of ∑ bixi) is not merely positive but is actually the same colour as thexi. Our main aim in this paper is to show that this sufficient condition is also necessary. We give a similar result when there is more than one constraint. As a special case, we determine which matricesBhave the following property: for every finite colouring of N there is a monochromatic vectorxwith all entries ofBxpositive.