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

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 $6\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/3\rfloor }\right)$. 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 $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/3\rfloor }\right)\cdot 2^{-\sqrt{16n/3}}$ for all sufficiently large $n$.

References in corpus (3)

• Progression-free sets in Z_4^n are exponentially small
• On cap sets and the group-theoretic approach to matrix multiplication
• Asymptotic upper bounds on progression-free sets in ${\mathbb{Z}}_p^n$

Cited by in corpus (1)

• A connection between matchings and removal in abelian groups