A framework for the generalised Erdős-Rothschild problem and a resolution of the dichromatic triangle case

TL;DR

This work presents a comprehensive framework that reduces the generalised Erdős-Rothschild problem to a colour-template optimisation problem $Q_t(\mathcal X)$, enabling asymptotic, stability, and exact results for families of forbidden colourings. It delivers two flagship results: (i) a complete solution for forbidding dichromatic triangles, showing extremal graphs are complete multipartite with an explicitly described set of active part-counts $R_2(s)$ and an asymptotic count of colourings, and (ii) a parallel result for forbidding improperly coloured cliques, outlining a similar phase-transition structure. The authors further develop a robust methodology combining analytic optimisation, regularity-based reductions, and stability to show that, under suitable conditions, extremal graphs are blow-ups of optimal templates and that almost all colourings follow these templates. Collectively, the results extend the understanding of general colour-pattern Erdős–Rothschild problems, establishing complete multipartite extremals for non-monochromatic patterns and proposing a blueprint for tackling further patterns via $Q_t(\mathcal X)$. The findings carry significance for combinatorial extremal counting, colour-pattern theory, and the broader study of phase transitions in extremal colouring problems.

Abstract

The Erdős-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 Turán 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 Erdős-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.

Figures (4)

• Figure 1.1: The pattern $K^{(2)}_3$ and the family $(K^{(2)}_3,4)$.
• Figure 2.1: A lower bound on the number of $K_3^{(2)}$-free $s$-edge colourings of $T_4(n)$ is given by splitting $[s]=C_1\cup C_2\cup C_3$ and colouring as shown.
• Figure 5.1: Plot (A) shows $\tilde{g}_s(x)$ for $s=300$ with maximum attained at $m(s)=W(s/e)+1$. Plot (B) displays the difference $e_s(r)=\tilde{g}_s(r)-g_s(r)$ for $s=300$. Note that the difference is $0$ whenever $s$ is divisible by $r-1$.
• Figure 5.2: Plot of $\tilde{f}_{s}(x)$ illustrating the hypothesis of \ref{['lm:fcomp']} for values $s=800$ and $r=5$ (noting that $\tilde{f}_{s}(x)$ and $f_{s}(x)$ are very close to each other).

Theorems & Definitions (88)

• Theorem 1.1: alon2004numberbalogh2006remarkbenevides2017edgebotler2019maximumkatherine_exact
• Theorem 2.1: hoppen2019remarkshoppen2022
• Definition 2.3: $g_s(r)$, $g(s)$, $R_2(s)$, $r_2(s)$
• Theorem 2.4
• Corollary 2.5
• Corollary 2.6
• Theorem 2.7
• proof
• proof
• Lemma 3.3