Ramsey functions involving Km,n with n large

For fixed integers m, k >= 2, it is shown that the k-color Ramsey number r(k) (K-m, (n)) and the bipartite Ramsey number b(k) (m, n) are both asymptotically equal to k(m)n as n -> infinity, and that for any graph H on m vertices, the two-color Ramsey number r (H + (K-n,) over bar K-n) is at most (1 + o(1))n(m+ 1) /(log n)(m-1). Moreover, the order of magnitude of r (H + (K-n,) over bar K-n) is proved to be n(m+) 1 / (log n)(m) if H not equal K-m as n -> infinity. (c) 2005 Elsevier B.V. All rights reserved.