Size Ramsey number of bipartite graphs and bipartite Ramanujan graphs

Given a graph $ G $, a graph $ F $ is said to be Ramsey for $ G $ if in every edge coloring of $F$ with two colors, there exists a monochromatic copy of $G$. The minimum number of edges of a graph $ F $ which is Ramsey for $ G $ is called the size-Ramsey number of $G$ and is denoted by $hat{r}(G)$. In 1983, Beck gave a linear upper bound (in terms of $n$) for $hat{r}(P_{n})$, where $ P_n $ is a path on $ n $ vertices, giving a positive answer to a question of ErdH{o}s. After that, different approaches were attempted by several authors to reduce the upper bound for $hat{r}(P_n)$ for sufficiently large $n$ and most of these approaches are based on the classic models of random graphs. Also, Haxell and Kohayakama in 1994 proved that the size Ramsey number of the cycle $ C_n $ is linear in terms $ n $, however the Szemeredi's regularity lemma is used in their proof and so no specific constant coefficient is provided. Here, we provide a method to obtain an upper bound for the size Ramsey number of a graph using good expander graphs such as Ramanujan graphs. In particular, we give an alternative proof for the linearity of the size Ramsey number of paths and cycles. Our method has two privileges in compare to the previous ones. Firstly, it proves the upper bound for every positive integer $n$ in comparison to the random graph methods which needs $ n $ to be sufficiently large. Also, due to the recent explicit constructions for bipartite Ramanujan graphs by Marcus, Spielman and Srivastava, we can constructively find the graphs with small sizes which are Ramsey for a given graph. We also obtain some results about the bipartite Ramsey numbers.