Ramsey numbers involving large dense graphs and bipartite Turán numbers

AbstractWe 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 (δ−o(1))n2/(logn)α edges as n→∞, then there exists a constant c=c(m,δ)>0 such thatr(F,G)⩾(c−o(1))n(logn)α+1(e(F)−1)/(m−2),where e(F) is the number of edges of F. We also show that for any fixed k⩾m⩾2,r(Km,k,Kn)⩽(k−1+o(1))nlognmas 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 thenex(F;N)⩾N2−1/s−εfor all sufficiently large N. This partially proves a conjecture of Erdős and Simonovits