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 A x = 0 has at least δ | G | m − k monochromatic solutions. Density versions of this counting result are also addressed.