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

Let G be a finite abelian group with exponent n, and 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 and |G| is large enough then, for each r-coloring of G∖{0}, there is δ depending only on r,n and m such that the homogeneous linear system Ax=0 has at least δ|G|m−k monochromatic solutions. Density versions of this counting result are also addressed.