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

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

2012-03-11

arxiv.org/abs/1203.2383v1

Mathematics

Combinatorics

24 pages

Scientific paper

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.

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