  The set-colouring Ramsey number $R_{r,s}(k)$ is defined to be the minimum $n$
such that if each edge of the complete graph $K_n$ is assigned a set of $s$
colours from $\{1,\ldots,r\}$, then one of the colours contains a monochromatic
clique of size $k$. The case $s = 1$ is the usual $r$-colour Ramsey number, and
the case $s = r - 1$ was studied by Erd\H{o}s, Hajnal and Rado in 1965, and by
Erd\H{o}s and Szemer\'edi in 1972.
  The first significant results for general $s$ were obtained only recently, by
Conlon, Fox, He, Mubayi, Suk and Verstra\"ete, who showed that $R_{r,s}(k) =
2^{\Theta(kr)}$ if $s/r$ is bounded away from $0$ and $1$. In the range $s = r
- o(r)$, however, their upper and lower bounds diverge significantly. In this
note we introduce a new (random) colouring, and use it to determine
$R_{r,s}(k)$ up to polylogarithmic factors in the exponent for essentially all
$r$, $s$ and $k$.
