We consider the following question of Bollobás: given an r-coloring of E(Kn), how large a k-connected subgraph can we find using at most s colors? 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(k2−k)r and (n−k+1)/(r−1)+r. The case s≥2 is considered in a subsequent paper (Liu et al.[6]), where we also discuss some of the more glaring open problems relating to this question.