Some Upper Bounds Of Ramsey Functions

This paper will devote to Graph Ramsey theory, besides introducing the development of Ramsey theory, recent situation, and fundamental theorems. Some open problems are also introduced here.In this paper, we study the size condition of any connected graph which is 3-good . A connected graph G of order n is called m-good if r(Km,G) = (m-!)(?1) + 1 . Let q denote the size of G. Burr, Erdos, Faudree, Rousseau and Schelp have shown that for all n>4, if 9<(17n + l)/15, then G is 3-good. We improve it as for all ?4,if q <(In-1)/6, then G is 3-good.Moreover, a new upper bound of r(Km,G) is given. It has beenproved by Burr, Erdos, Faudree, Rousseau and Schelp that for any connected (n.q) graph G, w>3, then r(Km,G) <(n + 2q)a , where . It is shown in this paper that r(Km,G)<[(1 + 2q)a], where. This is an improved bound for r(Km,G).2 There is also a new result on two bipartite graphs. Let H be a graphobtained from Km and Kn by joining them with a bridge, andm>n>a>b, then r(Kab,H) b , there is a constant o 0 such that|V(H)|+b-l