Maximum number of sum-free colorings in finite abelian groups

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 $q\equiv 2(\mathrm{mod}3)$. 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.