  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.
