A Ramsey-type theorem for the matching number regarding connected graphs
Abstract
A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter , every graph G with sufficiently large (G) contains a `well-structured' induced subgraph H with large (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.