An improvement on {\L}uczak's connected matchings method

TL;DR

This paper improves {}uczak's method for reducing problems about monochromatic paths and cycles in complete graphs to connected matchings, simplifying the determination of certain Ramsey numbers.

Contribution

It introduces a refined reduction technique that simplifies the analysis of monochromatic structures in complete graphs, enhancing previous methods.

Findings

Simplified proof for the 3-color Ramsey number of long paths.

New reduction method for monochromatic connected matchings.

Potential to extend to other monochromatic graph problems.

Abstract

A connected matching in a graph $G$ is a matching contained in a connected component of $G$. A well-known method due to {\L}uczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the $3$-colour Ramsey number of long paths, using a simpler argument than the original one by Gy\'arf\'as, Ruszink\'o, S\'ark\"ozy, and Szemer\'edi (2007).