A generalization of Ramsey theory for stars and one matching

A recent question in generalized Ramsey theory is that for fixed positive integers \(s\leq t\), at least how many vertices can be covered by the vertices of no more than \(s\) monochromatic members of the family \(\cal F\) in every edge coloring of \(K_n\) with \(t\) colors. This is related to {{\(d\)-chromatic Ramsey numbers}} introduced by Chung and Liu. In this paper, we first compute these numbers for stars generalizing the well-known result of Burr and Roberts. Then we extend a result of Cockayne and Lorimer to compute \(d\)-chromatic Ramsey numbers for stars and one matching.