Ramsey multiplicity and the Tur\'an coloring

TL;DR

This paper investigates the extremal properties of Turán colorings in two- and three-color edge colorings of complete graphs, establishing their optimality for certain graphs and advancing understanding of Ramsey multiplicity constants.

Contribution

It proves Turán colorings are extremal and unique for an infinite family of graphs, and determines the Ramsey multiplicity constant where the Burr--Rosta conjecture fails.

Findings

Turán coloring is extremal for an infinite family of graphs.

First to determine the Ramsey multiplicity constant where Burr--Rosta fails.

Conditional three-color result based on a conjecture on Ramsey numbers.

Abstract

Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed graph $H$. This conjecture was disproved independently by Sidorenko and Thomason. The first author later found quantitatively stronger counterexamples, using the Tur\'an coloring, in which one of the two colors spans a balanced complete multipartite graph. We prove that the Tur\'an coloring is extremal for an infinite family of graphs, and that it is the unique extremal coloring. This yields the first determination of the Ramsey multiplicity constant of a graph for which the Burr--Rosta conjecture fails. We also prove an analogous three-color result. In this case, our result is conditional on a certain natural conjecture on the behavior of two-color Ramsey numbers.