Rainbow triangles in three-colored graphs

Abstract

Erdős and Sós 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)+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=4k 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.