Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers Original Research Article

The Ramsey number r(H,Kn) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K2,m,Kn)⩽(m−1+o(1))(n/log n)2 and r(C2m,Kn)⩽c(n/log n)m/(m−1) for m fixed and n→∞. Also r(K2,n,Kn)=Θ(n3/log2 n) and r(C5,Kn)⩽cn3/2/log n.