Highly connected monochromatic subgraphs of multicolored graphs

We consider the following question of Bollobás: given an r-colouring of E(Kn), how large a k-connected subgraph can we find using at most s colours? We provide a partial solution to this problem when s = 1 (and n is not too small), showing that when r = 2 the answer is n−2k+2, when r = 3 the answer is ⌊n−k 2 ⌋+1 or ⌈n−k 2 ⌉ + 1, and when r − 1 is a prime power then the answer lies between n r−1 − 11(k − k)r and n−k+1 r−1 + r. The case s > 2 is considered in a subsequent paper [6], where we also discuss some of the more glaring open problems relating to this question.