Rainbow numbers for matchings and complete graphs

For a given graph H and n>=1, let f(n,H) denote the maximum number m for which it is possible to colour the edges of the complete graph K"n with m colours in such a way that each subgraph H in K"n has at least two edges of the same colour. Equivalently, any edge-colouring of K"n with at least rb(n,H)=f(n,H)+1 colours contains a rainbow copy of H. [email protected]?s, Simonovits and Sos have determined rb(n,K"k) for large enough n. Moreover, for k=3, they have shown that rb(n,K"3)=n. In this paper we will determine the rainbow numbers rb(n,K"k) for all n>=k>=4, and the rainbow numbers rb(n,kK"2) for all k>=2 and n>=3k+3.