Rainbow triangles in three-colored graphs

TL;DR

This paper proves a conjecture about the maximum number of rainbow triangles in three-colored complete graphs for large n and specific cases, using flag algebras and stability methods, and determines the limiting behavior.

Contribution

It confirms the conjectured recurrence relation for large n and for n divisible by 4, establishing the limit of the maximum rainbow triangles ratio and identifying the unique extremal structure.

Findings

Proved the conjecture for sufficiently large n.

Confirmed the conjecture for n divisible by 4.

Determined the limit of rainbow triangles ratio as n grows large.

Abstract

Erdos and Sos 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 = 4k for all k. These results imply that lim F(n) n^3/6 = 0.4, and determine the unique limit object. In the proof we use flag algebras combined with stability arguments.