The growth rate of tri-colored sum-free sets

Let G be an abelian group. A tri-colored sum-free set in G is a collection of triples (a(i), b(i), c(i)) in G such that a(i) + b(j) + 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 + . . . + 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.