  For $n\geq s> r\geq 1$ and $k\geq 2$, write $n \rightarrow (s)_{k}^r$ if
every hyperedge colouring with $k$ colours of the complete $r$-uniform
hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving
upon previous results by \textcite{AGLM14} and \textcite{EHMR84} we show that
\[ \text{if } r \geq 3 \text{ and } n \nrightarrow (s)_k^r \text{ then } 2^n
\nrightarrow (s+1)_{k+3}^{r+1}. \] This yields an improvement for some of the
known lower bounds on multicolour hypergraph Ramsey numbers.
  Given a hypergraph $H=(V,E)$, we consider the Ramsey-like problem of
colouring all $r$-subsets of $V$ such that no hyperedge of size $\geq r+1$ is
monochromatic. We provide upper and lower bounds on the number of colours
necessary in terms of the chromatic number $\chi(H)$. In particular we show
that this number is $O(\log^{(r-1)} (r \chi(H)) + r)$.
