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 $k\times m$ matrix with integer entries. We show that if $A$ satisfies some natural conditions and $|G|$ is large enough then, for each $r$--coloring of $G\setminus \{0\}$, there is $\delta$ depending only on $r,n$ and $m$ such that the homogeneous linear system $Ax=0$ has at least $\delta |G|^{m-k}$ monochromatic solutions. Density versions of this counting result are also addressed.