The Growth Rate of Tri-Colored Sum-Free Sets

Abstract

Let $G$ be an abelian group. A tri-colored sum-free set in $G^n$ is a collection of triples $({\mathbf{a}}_i,{\mathbf{b}}_i,{\mathbf{c}}_i)$ in $G^n$ such that ${\mathbf{a}}_i+{\mathbf{b}}_j+{\mathbf{c}}_k=0$ 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.

10 pages, published in Discrete Analysis

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Cited by in corpus (9)

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• The G-stable rank for tensors
• Universal points in the asymptotic spectrum of tensors
• Limits on the Universal Method for Matrix Multiplication
• A new upper bound for the size of a sunflower-free family
• Larger Corner-Free Sets from Combinatorial Degenerations