Ramsey Numbers of Connected Clique Matchings

TL;DR

This paper establishes the exact Ramsey number for connected clique matchings in 2-edge-coloured complete graphs, showing a precise threshold for the existence of a monochromatic connected subgraph with multiple disjoint cliques.

Contribution

The authors determine the exact Ramsey number for connected clique matchings, providing a tight bound that was previously unknown.

Findings

The Ramsey number for connected clique matchings is exactly (r^2 - r - 1)n - r + 1.

Below this number, such a monochromatic connected subgraph always exists.

The bound is tight and cannot be improved.

Abstract

We determine the Ramsey number of a connected clique matching. That is, we show that if $G$ is a $2$-edge-coloured complete graph on $(r^2 - r - 1)n - r + 1$ vertices, then there is a monochromatic connected subgraph containing $n$ disjoint copies of $K_r$, and that this number of vertices cannot be reduced.