The power of many colours

A classical problem, due to Gerencsér and Gyárfás from 1967, asks how large a monochromatic connected component can we guarantee in any r-edge colouring of K_n? We consider how big a connected component we can guarantee in any r-edge colouring of K_n if we allow ourselves to use up to s colours. This is actually an instance of a more general question of Bollobás from about 20 years ago which asks for a k-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided n is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form. We also consider a generalisation in a similar direction of a question first considered by ErdÅ's and Rényi in 1956, who considered given n and m, what is the smallest number of m-cliques which can cover all edges of K_n? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any r-edge colouring of K_n. We consider what happens if we allow ourselves to use up to s colours. We obtain a more complete understanding of the answer to this question for large n, in particular, determining it up to a constant factor for all 1 ≤s≤r, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.