On Ramsey numbers of complete graphs with dropped stars

Abstract

Let $r(G,H)$ be the smallest integer $N$ such that for any $2$-coloring (say, red and blue) of the edges of $K\_n$, $n\geqslant N$, there is either a red copy of $G$ or a blue copy of $H$. Let $K\_n-K\_{1,s}$ be the complete graph on $n$ vertices from which the edges of $K\_{1,s}$ are dropped. In this note we present exact values for $r(K\_m-K\_{1,1},K\_n-K\_{1,s})$ and new upper bounds for $r(K\_m,K\_n-K\_{1,s})$ in numerous cases. We also present some results for the Ramsey number of Wheels versus $K\_n-K\_{1,s}$.