Large stars with few colors

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 an old problem of Chung and Liu: for graph $G$ and integers $1\leq s