The Erd\H{o}s-Rothschild problem on edge-colourings with forbidden monochromatic cliques

Let $\mathbf{k}:=(k_1,\dots ,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\mathbf{k})$ 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 $F(n;\mathbf{k})$ to denote the maximum of $F(G;\mathbf{k})$ over all graphs $G$ on $n$ vertices. This problem was first considered by Erd\H{o}s and Rothschild in 1974, but it has been solved only for a very small number of non-trivial cases. We prove that, for every $\mathbf{k}$ and $n$, there is a complete multipartite graph $G$ on $n$ vertices with $F(G;$\mathbf{k}$) = F(n;$\mathbf{k}$)$\mathrm{.}Also,forevery$$\mathbf{k}$$ we construct a finite optimisation problem whose maximum is equal to the limit of $\log_2 F(n;$\mathbf{k}$)/{n\choose 2}$as$n$ tends to infinity. Our final result is a stability theorem for complete multipartite graphs $G$, describing the asymptotic structure of such $G$ with $F(G;$\mathbf{k}$) = F(n;$\mathbf{k}$) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem.