On the number of solutions of regular systems in sets with positive density in abelian groups

Abstract

We prove a density version for the Ramsey statement that, for fixed r and sufficiently large n, every coloring of an abelian group of order n with r colors has nontrivial monochromatic solutions of the system Ax = 0, where A is an integer matrix with row–sum zero. Our result shows that asymptotically the number of solutions is in fact as large as it can be expected.