Rainbow triangles in three-colored graphs

Erdős and Sós proposed a problem of determining the maximum number F (n) of rainbow triangles in 3-edge-colored complete graphs on n vertices. They conjectured that F (n) = F (a)+ F (b) +F (c) +F (d) + abc+ abd+ acd+ bcd, where a+ b+ c+ d = n and a, b, c, d are as equal as possible. We prove that the conjectured recurrence holds for sufficiently large n. We also prove the conjecture for n = 4 for all k ≥ 0. These results imply that lim F (n) (n3) = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.