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

TL;DR

This paper establishes a lower bound for the maximum size of k-colored sum-free sets in t^n_m, matching known upper bounds for prime power m, and generalizes previous results for k=3.

Contribution

It provides a new lower bound for sum-free sets in t^n_m that matches upper bounds when m is a prime power, extending prior work for k=3.

Findings

Lower bound matches upper bound for prime power m

Generalizes Kleinberg-Sawin-Speyer result for k=3

Extends Pebody's result with new ideas

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 $k\geq 3$ and $m\geq 2$ 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.