On a question of Erdos and Faudree on the size Ramsey numbers

Abstract

For given simple graphs $G_1$ and $G_2$, the size Ramsey number $\hat{R}(G_1,G_2)$ is the smallest positive integer $m$, where there exists a graph $G$ with $m$ edges such that in any edge coloring of $G$ with two colors red and blue, there is either a red copy of $G_1$ or a blue copy of $G_2$. In 1981, Erd\H{o}s and Faudree investigated the size Ramsey number $\hat{R}(K_n,tK_2)$, where $K_n$ is a complete graph on $n$ vertices and $tK_2$ is a matching of size $t$. They obtained the value of $\hat{R}(K_n,tK_2)$ when $n\geq 4t-1$ as well as for $t=2$ and asked for the behavior of these numbers when $ t $ is much larger than $ n $. In this regard, they posed the following interesting question: For every positive integer $n$, is it true that $$\lim_{t\to \infty} \frac{\hat{R}(K_n,tK_2)} {t\, \hat{R}(K_n,K_2)} = \min\left\{\dfrac{\binom{n+2t-2}{2}} {t\binom{n}{2}}\mid t\in \mathbb{N}\right\} ? $$ In this paper, we obtain the exact value of $ \hat{R}(K_n,tK_2) $ for every positive integers $ n,t $ and as a byproduct, we give an affirmative answer to the question of Erd\H{o}s and Faudree.