Multicolour containers and the entropy of decorated graph limits

In recent breakthrough results, Saxton–Thomason and Balogh–Morris–Samotij have developed powerful theories of hypergraph containers. These theories have led to a large number of new results on transference, and on counting and characterising typical graphs in hereditary properties. In a different direction, Hatami–Janson–Szegedy proved results on the entropy of graph limits which enable us to count and characterise graphs in dense hereditary properties. In this paper, we make a threefold contribution to this area of research: (i) We generalise results of Saxton–Thomason to obtain container theorems for general, dense hereditary properties of multicoloured graphs. Our main tool is the adoption of an entropybased framework. As corollaries, we obtain general counting, characterization and transference results. We further give a streamlined extension of our results to cover a great variety of combinatorial structures: directed graphs, oriented graphs, tournaments, multipartite graphs, multi-graphs, hypercubes and hypergraphs. (ii) We generalise the results of Hatami–Janson–Szegedy on the entropy of graph limits to the setting of decorated graph limits. In particular we define a cut norm for decorated graph limits and prove compactness of the space of decorated graph limits under that norm. (iii) We explore a weak equivalence between the container and graph limit approaches to counting and characterising graphs in hereditary properties. In one direction, we show how our multicolour containers may be used to fully recover decorated versions of the results of Hatami–Janson–Szegedy. In the other direction, we show that our decorated extensions of Hatami–Janson–Szegedy’s results on graph limits imply counting and characterization applications. Finally, we raise the problem of determining the possible structure of entropy maximisers in a multicoloured setting, and discuss the contrasts between the container and the graph limit approaches to counting. Similar container results were recently obtained independently by Terry.