  Let $\textbf{k} := (k_1,\ldots,k_s)$ be a sequence of natural numbers. For a
graph $G$, let $F(G;\textbf{k})$ denote the number of colourings of the edges
of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the
edges of colour $c$ contain no clique of order $k_c$. Write $F(n;\textbf{k})$
to denote the maximum of $F(G;\textbf{k})$ over all graphs $G$ on $n$ vertices.
There are currently very few known exact (or asymptotic) results for this
problem, posed by Erd\H{o}s and Rothschild in 1974. We prove some new exact
results for $n \to \infty$:
  (i) A sufficient condition on $\textbf{k}$ which guarantees that every
extremal graph is a complete multipartite graph, which systematically recovers
all existing exact results.
  (ii) Addressing the original question of Erd\H{o}s and Rothschild, in the
case $\textbf{k}=(3,\ldots,3)$ of length $7$, the unique extremal graph is the
complete balanced $8$-partite graph, with colourings coming from Hadamard
matrices of order $8$.
  (iii) In the case $\textbf{k}=(k+1,k)$, for which the sufficient condition in
(i) does not hold, for $3 \leq k \leq 10$, the unique extremal graph is
complete $k$-partite with one part of size less than $k$ and the other parts as
equal in size as possible.
