Multicolour containers, extremal entropy and counting

TL;DR

This paper applies hypergraph container theories and entropy methods to derive counting and stability results for hereditary properties of multicolourings across various combinatorial structures, unifying and extending previous work.

Contribution

It introduces a unified framework for container and counting theorems applicable to a wide range of coloured combinatorial objects, including new structures not previously studied with containers.

Findings

Derived container and counting theorems for hereditary properties of k-colourings.

Characterized stability and extremal entropy for sequences of complete graphs.

Extended the container method to directed graphs, tournaments, multigraphs, and multicoloured graphs.

Abstract

In breakthrough results, Saxton-Thomason and Balogh-Morris-Samotij developed powerful theories of hypergraph containers. In this paper, we explore some consequences of these theories. We use a simple container theorem of Saxton-Thomason and an entropy-based framework to deduce container and counting theorems for hereditary properties of k-colourings of very general objects, which include both vertex- and edge-colourings of general hypergraph sequences as special cases. In the case of sequences of complete graphs, we further derive characterisation and transference results for hereditary properties in terms of their stability families and extremal entropy. This covers within a unified framework a great variety of combinatorial structures, some of which had not previously been studied via containers: directed graphs, oriented graphs, tournaments, multigraphs with bounded multiplicity and multicoloured graphs amongst others. Similar results were recently and independently obtained by Terry.