Highly connected coloured subgraphs via the regularity lemma

For integers n, r, s, k ∈ N, n ≥ k and r ≥ s, let m(n, r, s, k) be the largest (in order) k-connected component with at most s colours one can find in any r-colouring of the edges of the complete graph Kn on n vertices. Bollobás asked for the determination of m(n, r, s, k). Here, bounds are obtained in the cases s = 1, 2 and k = o(n), which extend results of Liu, Morris and Prince. Our techniques use Szemerédi’s Regularity Lemma for many colours. We shall also study a similar question for bipartite graphs.