A nearly tight upper bound on tri-colored sum-free sets in characteristic 2

TL;DR

This paper establishes a nearly tight upper bound on the size of tri-colored sum-free sets in characteristic 2, using a variant of the Croot-Lev-Pach lemma, and demonstrates the bound's near optimality with explicit constructions.

Contribution

It introduces a new upper bound for tri-colored sum-free sets in F_2^n and proves its tightness up to a subexponential factor, advancing understanding of sum-free configurations.

Findings

Upper bound of $6 {n race loor{n/3}}$ on sum-free sets in $F_2^n$

Existence of sum-free sets larger than ${n race loor{n/3}} imes 2^{- ext{sqrt}(16n/3)}$

Bound is nearly tight, differing by a subexponential factor

Abstract

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, $\{(a_i,b_i,c_i)\}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by Croot, Lev, and Pach in their breakthrough work on arithmetic-progression-free sets, we prove that the size of any tri-colored sum-free set in $\mathbb{F}_2^n$ is bounded above by $6 {n \choose \lfloor n/3 \rfloor}$. This upper bound is tight, up to a factor subexponential in $n$: there exist tri-colored sum-free sets in $\mathbb{F}_2^n$ of size greater than ${n \choose \lfloor n/3 \rfloor} \cdot 2^{-\sqrt{16 n / 3}}$ for all sufficiently large $n$.