Rainbow triangles in three-colored graphs
Jozsef Balogh, Ping Hu, Bernard Lidicky, Florian Pfender, Jan Volec,, Michael Young

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.
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