his thesis deals with Ramsey numbers and size Ramsey numbers . To study size Ramsey numbers , random graphs, random regular graphs and random bipartite graphs play important roles . By finding appropriate r andom graphs and random regular graphs , we can find some linear upper bounds for size Ramsey numbers of sparse graphs especially cycles. In this thesis we first focus on the size Ramsey numbers of $ \\\\mathcal{ F} $ and $ H $ where $\\\\mathcal{ F} = \\\\mathcal{C} _{ \\\\leq cn} $ , is the family of cycles of length at most $ cn $ , and $ H = P_{n} $ . Using similar techniques, we also managed to analyze $ \\\\hat{r}(C_n, P_n) $ , which was investigated previously only by using the regularity method. In particular, we give various linear upper bounds for the size Ramsey numbers of cycles. \\\\\\\\ In ???? , Dudek, Pra?at gave a simple (probabilistic) proof showing that $ \\\\hat{r}(P_{n}) \\\\leq ???n $. The standard techniques for proving a linear bound for paths, without the use of the regularity lemma, would not have sufficed to prove a linear bound for cycles. In fact we show that the $ r-$ color size Ramsey number of a cycle of order $ n $ is linear in $ n $. This has already been proved by Haxell, Kohayakawa and {\\\\L}uczak (actually they proved that the induced size Ramsey number of $ C_{n} $ is linear in $ n $ ), but this new proof avoid s the use of the regularity lemma, thus we enable to give specific constants $ c_{k} $ , such that the $ k-$ color size Ramsey number of $ C_{n} $ is at most $ c_{k} \\\imes n $. on the other words, we show that $ c_{k} $ can be taken to be doubly-exponential in $ k $ , or, if $ n $ is even , $ c_{k} $ can be taken to be exponential in $ k $. کلیدواژه فارسی
