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 Kn with m colours in such a way that each subgraph H in Kn has at least two edges of the same colour. Equivalently, any edge-colouring of Kn with at least rb(n; H) = f(n; H) + 1 colours contains a rainbow copy of H . Erdős, Simonovits and S6 os have determined rb(n; Kk) for large enough n. Moreover, for k = 3, they have shown that rb(n; K3) = n. In this paper we will determine the rainbow numbers rb(n; Kk) for all n¿ k¿ 4, and the rainbow numbers rb(n; kK2) for all k¿ 2 and n¿ 3k + 3. c © 2004 Published by Elsevier B.V. MSC: 05C15; 05C35; 05C55