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

A major line of research is discovering Ramsey-type theorems, which are results of the following form: given a graph parameter ρ, every graph G with sufficiently large ρ(G) contains a particular induced subgraph H with large ρ(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, namely, every sufficiently large connected graph contains a large induced connected graph that is a complete graph, a large star, or a path. Given a graph G, the matching number and the induced matching number of G are 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.