A Ramsey-type theorem for the matching number regarding connected graphs

TL;DR

This paper establishes new Ramsey-type theorems for the matching number, induced matching number, and independence number in connected graphs, revealing structured subgraphs in large graphs based on these parameters.

Contribution

It introduces novel Ramsey-type theorems specifically for the matching number and induced matching number in connected graphs, expanding classical results.

Findings

Ramsey-type theorem for the matching number in connected graphs

Ramsey-type theorem for the induced matching number in connected graphs

Additional results for the independence number in connected graphs

Abstract

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter $\rho$, every graph $G$ with sufficiently large $\rho(G)$ contains a `well-structured' induced subgraph $H$ with large $\rho(H)$. The classical Ramsey's theorem deals with the case when the graph parameter under consideration is the number of vertices; there is also a Ramsey-type theorem regarding connected graphs. Given a graph $G$, the matching number and the induced matching number of $G$ is the maximum size of a matching and an induced matching, respectively, of $G$. In this paper, we formulate Ramsey-type theorems for the matching number and the induced matching number regarding connected graphs. Along the way, we obtain a Ramsey-type theorem for the independence number regarding connected graphs as well.