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

TL;DR

This paper characterizes the maximum number of edge-colourings avoiding monochromatic cliques of specified sizes, showing that extremal graphs are complete multipartite and linking the problem to an asymptotic optimization framework.

Contribution

It proves that extremal graphs are complete multipartite and establishes a connection between the problem and a finite optimization problem for asymptotic analysis.

Findings

Extremal graphs are complete multipartite.

A finite optimization problem characterizes the asymptotic maximum.

Stability results describe the structure of near-extremal graphs.

Abstract

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.