Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

Abstract

The Ramsey number r(H,K_n) 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 $\text{r(K}{}_{\mathrm{2,m}}\text{,K}{}_{\mathrm{n}}\text{)⩽(m−1+}\text{o}\text{(1))(n/}\text{log}\text{n)}{}^2$ and $\text{r(C}{}_{\mathrm{2m}}\text{,K}{}_{\mathrm{n}}\text{)⩽c(n/}\text{log}\text{n)}{}^{\mathrm{m/(m-1)}}$ for m fixed and n→∞. Also $\text{r(K}{}_{\mathrm{2,n}}\text{,K}{}_{\mathrm{n}}\text{)=Θ(n}{}^3\text{/}\text{log}{}^2\text{n)}$ and $\text{r(C}{}_5\text{,K}{}_{\mathrm{n}}\text{)⩽cn}{}^{\mathrm{3/2}}\text{/}\text{log}\text{n}$.