Erdős–Sós extremal coloring conjecture for rainbow triangles

Let $K_n$ be the complete graph on $n$ vertices with its edges colored using three colors. Let $F(G)$ be the number of rainbow triangles in a coloring $G$, and let

$$F(n)=\max_{G:\,\text{$3$-edge-colored }K_n}F(G).$$

A coloring is attained via Construction when it is obtained by fixing the unique, up to isomorphism, three-edge-coloring of $K_4$ in which every triangle is rainbow, blowing up its four vertices into classes of sizes $a,b,c,d$, inheriting the interclass edge colors, and placing extremal colorings inside the four classes. Erdős–Sós extremal coloring conjecture. For every $n$, all three-edge-colorings of $K_n$ attaining $F(n)$ are attained via this construction.

The construction gives the recursive formula conjectured by Erdős and Sós for the extremal number of rainbow triangles, with $a+b+c+d=n$ and $a,b,c,d$ as equal as possible. The paper proves that this recurrence holds for sufficiently large $n$ and for $n=4^k$, but the stronger assertion that every extremal coloring arises from the construction is not resolved in the supplied text.