多色容器与装饰图极限的熵
Multicolour containers and the entropy of decorated graph limits
摘要：
在最近的突破性成果中，Saxton--Mesason和Balogh--Morris--Samotij发展了超图容器的强大理论。这些理论导致了大量的新结果的转移，并计数和表征典型的图的遗传性质。在一个不同的方向，哈塔米-詹森-Szegedy证明结果熵的图限制计数和图的密集遗传性质。在本文中，我们在这一领域的研究作出了三方面的贡献：1）我们推广了Saxton--Kesason的结果，得到了多色图的一般稠密遗传性质的容器定理。我们的主要工具是采用基于熵的框架。作为推论，我们得到了一般的计数、刻画和迁移结果。我们进一步扩展我们的结果，以涵盖各种组合结构：有向图，定向图，比赛，多部图，多重图，超立方体和超图。2)我们将Hatami-Janson-Szegedy关于图极限的熵的结果推广到装饰图极限的设定上。特别是，我们定义了一个切割规范的装饰图的限制，并证明了紧的空间装饰图的限制下，规范。3)我们探讨了容器和图极限方法之间的弱等价性，以计数和表征遗传性质的图。在一个方向，我们展示了我们的多色容器的结果可以用来恢复装饰版本的结果Hatami-詹森-Szegedy。在另一个方向，我们表明，我们的装饰扩展Hatami-詹森-Szegedy的结果图极限意味着计数和表征应用。特里最近独立获得了类似的容器结果。
英文摘要：
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 count and characterise graphs in dense hereditary properties. In this paper, we make a threefold contribution to this area of research: 1) 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 entropy-based framework. As corollaries, we obtain general counting, characterization and transference results. We further extend our results to cover a variety of combinatorial structures: directed graphs, oriented graphs, tournaments, multipartite graphs, multi-graphs, hypercubes and hypergraphs. 2) 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. 3) 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 container results may be used to 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. Similar container results were recently obtained independently by Terry.
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