Size Ramsey numbers of stars versus cliques

TL;DR

This paper investigates the size Ramsey numbers of stars versus cliques, confirming a conjecture for large parameters and providing new bounds that extend previous results in graph Ramsey theory.

Contribution

The paper proves Faudree and Sheehan's conjecture for size Ramsey numbers of stars versus cliques when n is sufficiently large relative to k.

Findings

Confirmed the conjecture for n ≥ k^3 + 2k^2 + 2k.

Extended the understanding of size Ramsey numbers for large n.

Disproved the conjecture for small n in previous work.

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 $.