A nearly tight upper bound on tri-colored sum-free sets in characteristic 2

A tri-colored sum-free set in an abelian group $H$ is a collection of ordered triples in $H^3$, ${(a_i,b_i,c_i)}_{i=1}^m$, such that the equation $a_i+b_j+c_k=0$ holds if and only if $i=j=k$. Using a variant of the lemma introduced by Croot, Lev, and Pach in their breakthrough work on arithmetic-progression-free sets, we prove that the size of any tri-colored sum-free set in $mathbb{F}_2^n$ is bounded above by $6 {n choose lfloor n/3 rfloor}$. This upper bound is tight, up to a factor subexponential in $n$: there exist tri-colored sum-free sets in $mathbb{F}_2^n$ of size greater than ${n choose lfloor n/3 rfloor} cdot 2^{-sqrt{16 n / 3}}$ for all sufficiently large $n$.