Graphs with a number of edge roughly quadratic in their number of vertices are usually called dense . The number = roportion of its potential edge that G actually has is the edge density of G . The Ramsey number of a simple graph G is the least integer n such that every 2-edge-colouring of K n yields a monochromatic copy of G . This number is denoted by R(G,G) or R(G) . One of the oldest results in Ramsey Theory , proved by Erdos and Szekeres in 1935 , asserts that the Ramsey number of the complete graph with m edges is at most . we give a result due to Conlon which shows that for a given graph G , R(G)