  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 $3 \theta^n$. 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.
