A framework for the generalised Erd\H{o}s-Rothschild problem and a resolution of the dichromatic triangle case

TL;DR

This paper generalizes the Erd ext{o}s-Rothschild problem to forbidden families of colourings, providing a framework and solutions for specific cases, and characterizes extremal graphs as complete partite for all non-monochromatic patterns.

Contribution

It extends the Erd ext{o}s-Rothschild problem framework to broader forbidden colouring families and fully solves cases involving triangles with two colours and improperly coloured cliques.

Findings

Identifies extremal structures for specific forbidden families.

Shows extremal graphs are complete partite for all non-monochromatic patterns.

Provides an infinite family of extremal structures for certain cases.

Abstract

The Erd\H{o}s-Rothschild problem from 1974 asks for the maximum number of $s$-edge colourings in an $n$-vertex graph which avoid a monochromatic copy of $K_k$, given positive integers $n,s,k$. In this paper, we systematically study the generalisation of this problem to a given forbidden family of colourings of $K_k$. This problem typically exhibits a dichotomy whereby for some values of $s$, the extremal graph is the `trivial' one, namely the Tur\'an graph on $k-1$ parts, with no copies of $K_k$; while for others, this graph is no longer extremal and determining the extremal graph becomes much harder. We generalise a framework developed for the monochromatic Erd\H{o}s-Rothschild problem to the general setting and work in this framework to obtain our main results, which concern two specific forbidden families: triangles with exactly two colours, and improperly coloured cliques. We essentially solve these problems fully for all integers $s \geq 2$ and large $n$. In both cases we obtain an infinite family of structures which are extremal for some $s$, which are the first results of this kind. A consequence of our results is that for every non-monochromatic colour pattern, every extremal graph is complete partite. Our work extends work of Hoppen, Lefmann and Schmidt and of Benevides, Hoppen and Sampaio.