Maximum number of sum-free colorings in finite abelian groups

TL;DR

This paper determines the maximum number of sum-free colorings in subsets of finite abelian groups, revealing a strong link to the structure of largest sum-free sets and extending known results using the container method.

Contribution

It characterizes when subsets achieve maximum sum-free colorings in large abelian groups, especially relating to the uniqueness and union of largest sum-free sets.

Findings

For groups of type I, maximum colorings occur in largest sum-free sets.

For even order groups, the phenomenon extends to unions of largest sum-free sets.

The container method is used to extend results to larger numbers of colors.

Abstract

An $r$-coloring of a subset $A$ of a finite abelian group $G$ is called sum-free if it does not induce a monochromatic Schur triple, i.e., a triple of elements $a,b,c\in A$ with $a+b=c$. We investigate $\kappa_{r,G}$, the maximum number of sum-free $r$-colorings admitted by subsets of $G$, and our results show a close relationship between $\kappa_{r,G}$ and largest sum-free sets of $G$. Given a sufficiently large abelian group $G$ of type $I$, i.e., $|G|$ has a prime divisor $q$ with $q\equiv 2\pmod 3$. For $r=2,3$ we show that a subset $A\subset G$ achieves $\kappa_{r,G}$ if and only if $A$ is a largest sum-free set of $G$. For even order $G$ the result extends to $r=4,5$, where the phenomenon persists only if $G$ has a unique largest sum-free set. On the contrary, if the largest sum-free set in $G$ is not unique then $A$ attains $\kappa_{r,G}$ if and only if it is the union of two largest sum-free sets (in case $r=4$) and the union of three ("independent") largest sum-free sets (in case $r=5$). Our approach relies on the so called container method and can be extended to larger $r$ in case $G$ is of even order and contains sufficiently many largest sum-free sets.