On the Ramsey numbers $r(S_n,K_6-3K_2)$

Abstract

For every connected graph $F$ with $n$ vertices and every graph $G$ with chromatic surplus $s(G)\le n$, the Ramsey number $r(F,G)$ satisfies $r(F,G)\ge (n-1)(\chi (G)-1)+s(G),$ where $\chi (G)$ denotes the chromatic number of $G$. If this lower bound is attained, then $F$ is called $G$-good. For all connected graphs $G$ with at most six vertices and $\chi (G)\ge 4$, every tree $T_n$ of order $n\ge 5$ is $G$-good. In case of $\chi (G)=3$ and $G\ne K_6-3K_2$, every non-star tree $T_n$ is $G$-good except for some small $n$, whereas $r(S_n,G)$ for the star $S_n=K_{1,n-1}$ in a few cases differs by at most 2 from the lower bound. In this note, we prove that the values of $r(S_n,K_6-3K_2)$ are considerably larger for sufficiently large $n$. Furthermore, exact values of $r(S_n,K_6-3K_2)$ are obtained for small $n$.