A generalization of Ramsey theory for stars and one matching

A recent question in generalized Ramsey theory is that for fixed positive integers $s\le t$, at least how many vertices can be covered by the vertices of no more than $s$ monochromatic members of the family $\mathcal{F}$ 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.