New Ramsey Multiplicity Bounds and Search Heuristics

TL;DR

This paper advances bounds on Ramsey multiplicities for small cliques, introduces off-diagonal variants, and uses computational methods to identify extremal graph constructions, contributing to understanding graph density limits.

Contribution

It improves upper bounds on Ramsey multiplicities for K4 and K5, introduces off-diagonal variants, and employs search heuristics with flag algebras for extremal graph construction.

Findings

Improved upper bounds on Ramsey multiplicity of K4 and K5.

Identified extremal constructions as blow-ups of small graphs.

Established stability and density region results for graph limits.

Abstract

We study two related problems concerning the number of homogeneous subsets of given size in graphs that go back to questions of Erd\H{o}s. Most notably, we improve the upper bounds on the Ramsey multiplicity of $K_4$ and $K_5$ and settle the minimum number of independent sets of size $4$ in graphs with clique number at most $4$. Motivated by the elusiveness of the symmetric Ramsey multiplicity problem, we also introduce an off-diagonal variant and obtain tight results when counting monochromatic $K_4$ or $K_5$ in only one of the colors and triangles in the other. The extremal constructions for each problem turn out to be blow-ups of a graph of constant size and were found through search heuristics. They are complemented by lower bounds established using flag algebras, resulting in a fully computer-assisted approach. For some of our theorems we can also derive that the extremal construction is stable in a very strong sense. More broadly, these problems lead us to the study of the region of possible pairs of clique and independent set densities that can be realized as the limit of some sequence of graphs.