Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a question of Erdos and Faudree on the size Ramsey numbers   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.

7
The categories of this paper are: hep-th
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
3
The categories of this paper are: physics
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin