Highly connected multicoloured subgraphs of multicoloured graphs

Suppose the edges of the complete graph on n vertices are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobas, and the case s = 1 was considered in a preivous paper by the same authors. Here we consider the case s >= 2, and mention some open problems.