Exact solutions to the Erdős-Rothschild problem

Abstract

Let $\text{k}:=(k_1,\dots ,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\text{k})$ 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 $F(n;\text{k})$ to denote the maximum of $F(G;\text{k})$ over all graphs $G$ on $n$ vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for $n\to \infty$: (i) A sufficient condition on $\text{k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results. (ii) Addressing the original question of Erdős and Rothschild, in the case $\text{k}=(3,\dots ,3)$ of length $7$, the unique extremal graph is the complete balanced $8$-partite graph, with colourings coming from Hadamard matrices of order $8$. (iii) In the case $\text{k}=(k+1,k)$, for which the sufficient condition in (i) does not hold, for $3\le k\le 10$, the unique extremal graph is complete $k$-partite with one part of size less than $k$ and the other parts as equal in size as possible.

arXiv comment54 pages, 2 figures, 5 ancillary programs. To appear in Forum of Math., Sigma