A very robust Ramsey theorem for matchings

TL;DR

This paper extends the multicolour Ramsey theorem for matchings to a broader class of graphs called s-connectors, showing they retain similar properties to complete graphs with only minor losses, using advanced combinatorial techniques.

Contribution

It introduces a robust generalisation of the Cockayne-Lorimer theorem applicable to s-connector graphs, broadening the understanding of Ramsey properties in sparse graphs.

Findings

s-connector graphs have similar Ramsey matching properties as complete graphs

The loss in Ramsey matching number is only an additive O(s)

The proof uses a novel compression algorithm based on Gallai-Edmonds decompositions

Abstract

Our main result is a robust generalisation of the Cockayne-Lorimer theorem on the multicolour Ramsey number of matchings. It is moreover a generalisation of the transference generalisation of Cockayne-Lorimer, which (informally) says that the random graph $G \sim G(n,p)$ with $np \to \infty$ has, with high probability, essentially the same Ramsey matching properties as the complete graph $K_n$. We show, somewhat surprisingly, that the same is true under the rather weak robustness assumption that $G$ is an $s$-connector (i.e. $\overline{G}$ is $K_{s,s}$-free) with $s=o(n)$. Moreover, we show that such $G$ has only an additive $O(s)$ loss with respect to $K_n$ for monochromatic matchings, which is essentially sharp. Our proof adapts a compression algorithm based on Gallai-Edmonds decompositions that we developed previously for generalised Ramsey-Tur\'an problems.