For given graphs H_1,…,H_t the multicolor size Ramsey number r ?(H_1,…,H_t) is the smallest integer m such that there exists a graph G on m edges such that in every t-edge coloring of G there is a monochromatic copy of H_i of color i for some 1?i?t. ) by Erd?s, Faudree, Rousseau and Schelp. Since then, the size Ramsey numbers of graphs have been studied with particular focus on the case of trees, bounded degree graphs and sparse graphs. Moreover, size Ramsey number of a sparse graph versus a dense graph has its special importance. Determining the exact value of r ?(K_(1,k) ,K_n) is one of the most studied directions in this area. In 1983, Faudree and Sheehan posed a conjecture that was an extension of a conjecture of Erd?s and determined the exact value of r ?(K_(1,k) ,K_n ). In the first part of this thesis, we study this conjecture and prove an important case of this conjecture. Moreover, we generalize our results on r ?(K_(1,k) ,K_n) for multi copies of stars versus multi copies of cliques. It is worth noting that determining the exact value of size Ramsey numbers for even a simple graph like a path P_n seems quite difficult. In 1983, answering a question posed by Erd?s, Beck proved that for sufficiently large n, we have r ?(P_n ,P_n )? 900n. Since then the study of the size Ramsey number of bounded degree graphs has been at the center of attention. Beck asked whether r ?(G ,G) grows linearly in the number of vertices for graphs of bounded degree. In fact, for the graph (n^(2-1/d #43;o(1) ) ). In the second part of this thesis, we give an alternative proof for the linearity of r ?(P_n,P_n) and answer the conjecture of Colon and Nenadov for grid graphs.
