The $d$-chromatic Ramsey number for stars

In 1978, Chung and Liu generalized the definition of the Ramsey number. They introduced the $d$-chromatic Ramsey number as follows. Let $1\le d<c$ be integers and let $A_1,\dots ,A_t$ be subsets with size $d$ of $[c]$, where $t=\left(\genfrac{}{}{0px}{}{c}{d}\right)$. For given graphs $G_1,\dots ,G_t$, {\it the $d$-chromatic Ramsey number}, $r^{d,c}(G_1,\dots ,G_t)$, is the minimum positive integer $N$ such that every $c$-coloring of $E(K_N)$ yields a copy of $G_i$ whose edges are colored by colors in the color set $A_i$ for some $i\in [t]$. The {\it star-critical $d$-chromatic Ramsey number}, $r_{\ast }^{d,c}(G_1,\dots ,G_t)$, is the minimum positive integer $k$ such that every $c$-coloring of $E(K_N-K_{1,N-1-k})$ yields a copy of $G_i$ whose edges are colored by colors in the color set $A_i$ for some $i\in [t]$, where $N=r^{d,c}(G_1,\dots ,G_t)$. If $G_1,\dots ,G_t=G$, then we simplify them as $r^{d,c}(G)$ (it also call {\it the weakened Ramsey number}) and $r_{\ast }^{d,c}(G)$, respectively. In this paper, we determine all the value of $r^{d,c}(K_{1,n})$, $r_{\ast }^{d,c}(K_{1,n})$ and part of the value of $r^{d,c}(K_{1,n_1},\dots ,K_{1,n_t})$.