Highly Connected Monochromatic Subgraphs of Multicoloured Graphs
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.), where we also discuss some of the more glaring open problems relating to this question.
Liu, H., Morris, R., Prince, N. (2009). Highly connected monochromatic subgraphs of multicoloured graphs. Journal of Graph Theory, 61(1), 22-44.