The Growth Rate of Tri-Colored Sum-Free Sets
Discrete analysis (2018)
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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 $θ= \min_{ρ>0} (1+ρ+\cdots + ρ^{q-1}) ρ^{-(q-1)/3}$. 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 $3 θ^n$. Between this paper and a paper of Pebody, we will show that, for any $δ> 0$, and $n$ sufficiently large, there are tri-colored sum-free sets in $C_q^n$ of size $(θ-δ)^n$. Our construction also works when $q$ is not prime.
@article{DAS_2018_a7,
author = {Robert Kleinberg and Will Sawin and David E. Speyer},
title = {The {Growth} {Rate} of {Tri-Colored} {Sum-Free} {Sets}},
journal = {Discrete analysis},
year = {2018},
language = {en},
url = {http://geodesic.mathdoc.fr/item/DAS_2018_a7/}
}
Robert Kleinberg; Will Sawin; David E. Speyer. The Growth Rate of Tri-Colored Sum-Free Sets. Discrete analysis (2018). http://geodesic.mathdoc.fr/item/DAS_2018_a7/