The topics of this thesis lie in graph Ramsey theory. Given two graphs G and H, by the Ramsey theorem, there exist infinitely many graphs F such that if we partition the edges of F into two sets, say Red and Blue, then either the graph induced by the red edges contains G or the graph induced by the blue edges contains H. The minimum order of F is called the Ramsey number and the minimum of the size of F is called the size Ramsey number. They are denoted by r(G, H) and ˆr(G, H), respectively. We will investigate size Ramsey numbers involving double stars and brooms.