Multicolour containers, extremal entropy and counting
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.
Cite
@article{arxiv.1801.05195,
title = {Multicolour containers, extremal entropy and counting},
author = {Victor Falgas-Ravry and Kelly O'Connell and Andrew Uzzell},
journal= {arXiv preprint arXiv:1801.05195},
year = {2018}
}
Comments
47 pages. This paper replaces, improves and expands roughly the first half of the preprint "Multicolour containers and the entropy of decorated graph limits" by the authors and Johanna Str\"omberg. The second half of that paper, dealing with graph limits, will appear separately. arXiv admin note: text overlap with arXiv:1607.08152