An upper bound for the Ramsey numbers \(r(K_ 3,G)\) (Q1322269)

Given graphs \(H\) and \(G\), the Ramsey number \(r(H,G)\) is defined to be the smallest integer \(n\) so that, whenever the edges of the complete graph on \(n\) vertices \(K_ n\) is edge 2-colored, then either there is a copy of \(H\) in \(K_ n\) with all of its edges colored with the first color or a copy of \(G\) with all of its edges colored with the second color. This paper is devoted to proving the following result: For any graph \(G\) with \(q\) edges and without isolated vertices, \(r(K_ 3,G) \leq 2q + 1\). This result may be restated as follows: Any graph on \(2q+1\) vertices with independence number at most 2 contains every (isolated-free) graph on \(q\) edges.