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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 $3 \\theta^n$. 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Speyer, Robert Kleinberg, Will Sawin","submitted_at":"2016-06-30T21:04:11Z","abstract_excerpt":"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 $3 \\theta^n$. 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