Highly connected coloured subgraphs via the regularity lemma

For integers n,r,s,k∈Nn,r,s,k∈N, n≥kn≥k and r≥sr≥s, let m(n,r,s,k)m(n,r,s,k) be the largest (in order) kk-connected component with at most ss colours one can find in any  rr-colouring of the edges of the complete graph KnKn on nn vertices. Bollobás asked for the determination of m(n,r,s,k)m(n,r,s,k).Here, bounds are obtained in the cases s=1,2s=1,2 and k=o(n)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.