On the number of monochromatic solutions of integer linear systems on Abelian groups

Let G be a finite cyclic group an let r be a positive integer. Let A be a k × m matrix with integer entries. We show that, if A satisfies some natural conditions, then the homogeneous linear system A x = 0 has Ω ( | G N | m − k ) monochromatic solutions for each r-coloring of G N \ { 0 } and sufficiently large N. Density versions of this counting result are also addressed.