Ramsey numbers involving large dense graphs and bipartite Turan numbers

We prove that for any fixed integer m greater than or equal to 3 and constants delta > 0 and alpha greater than or equal to 0, if F is a graph on m vertices and G is a graph on n vertices that contains at least (delta - o(1))n(2)/(log)(alpha) edges as n --> infinity, then there exists a constant c = c(m, delta) > 0 such that r(F, G) greater than or equal to (c - o(1)) (n/(log n)(alpha+1)) ((c(F)-1)/(m-2)), where e(F) is the number of edges of F. We also show that for any fixed k greater than or equal to m greater than or equal to 2 r(K(m,k), K(n)) less than or equal to (k - 1 + o (1)) (n/log n)(m) as n --> infinity. In addition, we establish the following result: For an m x k bipartite graph F with minimum degree s and for any epsilon > 0, if k > m/e then ex(F; N) greater than or equal to N(2-1/s-epsilon) for all sufficiently large N. This partially proves a conjecture of Erdos and Simonovits. (C) 2002 Elsevier Science (USA). All rights reserved.