Ramsey Numbers of Fans and Large Books

Abstract

For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $G$ or a blue $H$. Let $G+H$ be the graph obtained from vertex disjoint $G$ and $H$ by adding new edges connecting $G$ and $H$ completely, $F_m=K_1+mK_2$ and $B_p(n)=K_p+nK_1$. It is shown $R(F_m,B_p(n))=2(n+p-1)+1$ for fixed $m, p$ and large $n$.