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 $F(G;\mathbf{ k}) = F(n;\mathbf{ k})$. Also, for every $\mathbf{ k}$ 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.
Categories: math.CO