The power of many colours

Abstract: A classical problem, due to Gerencs\'er and Gy\'arf\'as 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 can we 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 Bollob\'as 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 Erd\H{o}s and R\'enyi 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 $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.