Ramsey numbers involving large dense graphs and bipartite Turán numbers

Abstract

We prove that for any fixed integer m⩾3 and constants δ>0 and α⩾0, if F is a graph on m vertices and G is a graph on n vertices that contains at least $\text{(δ−o(1))n}{}^2\text{/(}\text{log}\text{n)}{}^{\mathrm{\alpha }}$ edges as n→∞, then there exists a constant c=c(m,δ)>0 such that

$$\text{r(F,G)⩾(c−o(1))}\text{n}\text{(}\text{log}\text{n)}{}^{\mathrm{\alpha +1}}{}^{\mathrm{(e(F)-1)/(m-2)}}\text{,}$$

where e(F) is the number of edges of F. We also show that for any fixed k⩾m⩾2,

$$\text{r(K}{}_{\mathrm{m,k}}\text{,K}{}_{\mathrm{n}}\text{)⩽(k−1+o(1))}\text{n}\text{log}\text{n}{}^{\mathrm{m}}$$

as n→∞. In addition, we establish the following result: For an m×k bipartite graph F with minimum degree s and for any ε>0, if k>m/ε then

$$\text{ex(F;N)⩾N}{}^{\mathrm{2-1/s-\epsilon }}$$

for all sufficiently large N. This partially proves a conjecture of Erdős and Simonovits.