  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.
