  For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest
positive integer n such that each blue-red edge coloring of the complete graph
Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured
that there is an absolute constant c such that R(G) = R(G,G) < 2c p m for any
graph G with m edges and no isolated vertices. Recently this conjecture was
proved by B. Sudakov. In this note, using the Sudakovs ideas we give an
extension of his result and some interesting corollaries.
