The $s$-chromatic Ramsey number for stars

Abstract

In 1977, Chung, Chung and Liu generalized the definition of the Ramsey number. They introduced the $s$-chromatic Ramsey number as follows. Let $1\le s<t$ be integers and let $A_1,A_2,\dots ,A_c$ be subsets with size $s$ of $[t]$, where $c=\left(\genfrac{}{}{0ex}{}{t}{s}\right)$. For given graphs $G_1,G_2,\dots ,G_c$, the {\it $s$-chromatic Ramsey number} $r^{s,t}(G_1,G_2,\dots ,G_c)$, is the minimum positive integer $N$ such that every $t$-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 $i\in [c]$. The {\it star-critical $s$-chromatic Ramsey number} $r_{\ast }^{s,t}(G_1,G_2,\dots ,G_c)$, is the minimum integer $\ell$ such that every $t$-coloring of the edges in $K_N-E(K_{1,N-1-\ell })$ yields a copy of $G_i$ whose edges are colored by colors in the color set $A_i$ for some $i\in [c]$, where $N=r^{s,t}(G_1,G_2,\dots ,G_c)$. If $G_1=G_2=\cdots =G_c=G$, then we simplify them to $r^{s,t}(G)$ (also called the {\it weakened Ramsey number}) and $r_{\ast }^{s,t}(G)$, respectively. In this paper, we determine all the values of $r^{s,t}(K_{1,m})$ and $r_{\ast }^{s,t}(K_{1,m})$, and part of the value of $r^{s,t}(K_{1,m_1},K_{1,m_2},\dots ,K_{1,m_c})$.