Ramsey Numbers and the Size of Graphs

For two graph H and G, the Ramsey number r(H,G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph Kn on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdős and Harary, in this note we study how the Ramsey number r(Ks, G) depends on the size of the graph G. For s ≥ 3, we prove that for every G with m edges, r(Ks, G) ≥ c ( m/ logm ) s+1 s+3 for some positive constant c depending only on s. This lower bound improves an earlier result of Erdős, Faudree, Rousseau, and Schelp, and is tight up to a polylogarithmic factor when s = 3. We also study the maximum value of r(Ks, G) as a function of m.