Rainbow triangles in three-colored graphs

Abstract Erdős and Sos proposed the 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 ) + a b c + a b d + a c d + b c d , 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 k for all k ≥ 0 . These results imply that lim ⁡ F ( n ) ( n 3 ) = 0.4 , and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.