Size Ramsey numbers of stars versus cliques

Abstract

The size Ramsey number $ \hat{r}(G,H) $ of two graphs $ G $ and $ H $ is the smallest integer $ m $ such that there exists a graph $ F $ on $ m $ edges with the property that every red-blue colouring of the edges of $ F $, yields a red copy of $ G $ or a blue copy of $ H $. In $ 1981 $, Erd\H{o}s observed that $\hat{r}(K_{1,k},K_{3})\leq \binom{2k+1}{2}-\binom{k}{2}$ and he conjectured that the corresponding upper bound on $ \hat{r}(K_{1,k},K_{3}) $ is sharp. In $ 1983 $, Faudree and Sheehan extended this conjecture as follows: \hat{r}(K_{1,k},K_{n})=\left \{ {lr} \binom{k(n-1)+1}{2}-\binom{k}{2} & ~k\geq n~ \text{or}~ k~ \text{odd}. \binom{k(n-1)+1}{2}-k(n-1)/2 & \text{otherwise}. \right. They proved the case $ k=2 $. In $ 2001 $, Pikhurko showed that this conjecture is not true for $ n=3 $ and $ k\geq 5 $, disproving the mentioned conjecture of Erd\H{o}s. Here we prove Faudree and Sheehan's conjecture for a given $ k\geq 2 $ and $ n\geq k^{3}+2k^{2}+2k $.