Ramsey functions involving Km,n with n large

For fixed integers m,k⩾2, it is shown that the k-color Ramsey number rk(Km,n) and the bipartite Ramsey number bk(m,n) are both asymptotically equal to kmn as n→∞, and that for any graph H on m vertices, the two-color Ramsey number r(H+K¯n,Kn) is at most (1+o(1))nm+1/(logn)m-1. Moreover, the order of magnitude of r(H+K¯n,Kn) is proved to be nm+1/(logn)m if H≠Km as n→∞.