Let k:=(k1,…,ks) be a sequence of natural numbers. For a
graph G, let F(G;k) denote the number of colourings of the edges
of G with colours 1,…,s such that, for every c∈{1,…,s}, the
edges of colour c contain no clique of order kc. Write F(n;k)
to denote the maximum of F(G;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 k and n, there is a complete
multipartite graph G on n vertices with F(G;k)=F(n;k). Also, for every k we construct a finite
optimisation problem whose maximum is equal to the limit of log2F(n;k)/(2n) 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;k)=F(n;k)⋅2o(n2) in terms of solutions to the optimisation problem.Comment: 16 pages, to appear in Math. Proc. Cambridge Phil. So