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

TL;DR

This paper investigates the number of monochromatic solutions to linear systems over finite abelian groups under various colorings, establishing conditions for guaranteed solutions and their density versions.

Contribution

It provides new bounds and conditions ensuring a minimum number of monochromatic solutions for linear systems on large finite abelian groups.

Findings

Existence of a positive lower bound on monochromatic solutions for large groups.

Density versions of the monochromatic solution counts.

Results depend only on the number of colors, group exponent, and system size.

Abstract

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.