The Growth Rate of Tri-Colored Sum-Free Sets

TL;DR

This paper investigates the maximum size of tri-colored sum-free sets in abelian groups, establishing bounds that nearly match previous upper bounds and extending results to non-prime cyclic groups.

Contribution

It proves the existence of large tri-colored sum-free sets in $C_q^n$ approaching the known upper bounds, for any fixed $ heta$ and sufficiently large $n$, including non-prime $q$.

Findings

Tri-colored sum-free sets can be as large as $( heta - ext{small delta})^n$ for large $n$.

The construction extends to non-prime cyclic groups.

The results nearly match the upper bounds established by prior work.

Abstract

Let $G$ be an abelian group. A tri-colored sum-free set in $G^n$ is a collection of triples $({\bf a}_i, {\bf b}_i, {\bf c}_i)$ in $G^n$ such that ${\bf a}_i+{\bf b}_j+{\bf c}_k=0$ if and only if $i=j=k$. Fix a prime $q$ and let $C_q$ be the cyclic group of order $q$. Let $\theta = \min_{\rho>0} (1+\rho+\cdots + \rho^{q-1}) \rho^{-(q-1)/3}$. Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans (building on previous work of Croot, Lev and Pach, and of Ellenberg and Gijswijt) showed that a tri-colored sum-free set in $C_q^n$ has size at most $3 \theta^n$. Between this paper and a paper of Pebody, we will show that, for any $\delta > 0$, and $n$ sufficiently large, there are tri-colored sum-free sets in $C_q^n$ of size $(\theta-\delta)^n$. Our construction also works when $q$ is not prime.