A lower bound for the $k$-multicolored sum-free problem in ${\mathbb{Z}}_m^n$

Abstract

In this paper, we give a lower bound for the maximum size of a $k$-colored sum-free set in ${\mathbb{Z}}_m^n$, where $k\ge 3$ and $m\ge 2$ are fixed and $n$ tends to infinity. If $m$ is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a $k$-colored sum-free set in ${\mathbb{Z}}_m^n$. This generalizes a result of Kleinberg-Sawin-Speyer for the case $k=3$ and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg-Sawin-Speyer. Both of these generalizations require several key new ideas.