A conjecture of \(3\)-colored triangles in the complete graph

Conjecture (proposed by Erdös and Sós [1])

Suppose the edges of \(K_n\) are \(3\)-colored so that the number of \(3\)-colored triangles is maximized, say, having \(F(n)\) such triangles. Then \[ F(n)=F(a)+F(b)+F(c) +F(d) + (a^{-1}+b^{-1}+c^{-1}+d^{-1})abcd \] where \(a+b+c+d=n\) and \(a,b,c,d\) are as equal as possible.

Bibliography
1 J. Nešetril and V. Rödl, eds., Mathematics of Ramsey Theory, Springer-Verlag, Berlin, 1990.