Stability for the Erdős-Rothschild problem

Given a sequence $\boldsymbol {k} := (k_1,\ldots ,k_s)$ of natural numbers and a graph G, let $F(G;\boldsymbol {k})$ 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 $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of nontrivial cases. In previous work with Pikhurko and Yilma, (Math. Proc. Cambridge Phil. Soc. 163 (2017), 341–356), we constructed a finite optimisation problem whose maximum is equal to the limit of $\log _2 F(n;\boldsymbol {k})/{n\choose 2}$ as n tends to infinity and proved a stability theorem for complete multipartite graphs G. In this paper, we provide a sufficient condition on $\boldsymbol {k}$ which guarantees a general stability theorem for any graph G, describing the asymptotic structure of G on n vertices with $F(G;\boldsymbol {k}) = F(n;\boldsymbol {k}) \cdot 2^{o(n^2)}$ in terms of solutions to the optimisation problem. We apply our theorem to systematically recover existing stability results as well as all cases with $s=2$ . The proof uses a version of symmetrisation on edge-coloured weighted multigraphs.