Structure and enumeration theorems for hereditary properties in finite relational languages

Abstract

Given a finite relational language L, a hereditary L-property is a class of finite L-structures which is closed under isomorphism and model theoretic substructure. This notion encompasses many objects of study in extremal combinatorics, including (but not limited to) hereditary properties of graphs, hypergraphs, and oriented graphs. In this paper, we generalize certain definitions, tools, and results form the study of hereditary properties in combinatorics to the setting of hereditary L-properties, where L is any finite relational language with maximum arity at least two. In particular, the goal of this paper is to generalize how extremal results and stability theorems can be combined with standard techniques and tools to yield approximate enumeration and structure theorems. We accomplish this by generalizing the notions of extremal graphs, asymptotic density, and graph stability theorems using structures in an auxiliary language associated to a hereditary L-property. Given a hereditary L-property H, we prove an approximate asymptotic enumeration theorem for H in terms of its generalized asymptotic density. Further we prove an approximate structure theorem for H, under the assumption of that H has a stability theorem. The tools we use include a new application of the hypergraph containers theorem (Balogh-Morris-Samotij [14], ) to the setting of L-structures, a general supersaturation theorem for hereditary L-properties (also new), and a general graph removal lemma for L-structures proved by Aroskar and Cummings in [5].

Multicolor containers, extremal entropy, and counting

“…There is a significant overlap between the container, enumeration, and stability results presented here and those obtained independently by Terry [58] (see the discussion below), although the emphases and arguments of our two papers are quite different. Since the first version of this paper was written, Terry has further explored interesting related questions on the different possible speeds in general multicolor properties [59], going beyond the case of "dense" properties we consider in the present work.…”

“…As indicated above, the results of Sections 2.2‐2.5 are very similar to those recently obtained by Terry . In particular, Terry's Theorems 2, 3, 6, and 7 correspond to our Proposition , Corollary , Theorem , and Lemma , respectively, while Terry's Theorem 5 is very similar to our Theorem .…”

“…Finally in Sections 2.5 and 2.6 we obtain general characterization and transference results (Theorems 2.18 and 2.23, respectively). As indicated above, the results of Sections 2.2-2.5 are very similar to those recently obtained by Terry [58]. In particular, Terry's Theorems 2, 3, 6, and 7 correspond to our Proposition 2.10, Corollary 2.15, Theorem 2.6, and Lemma 2.11, respectively, while Terry's Theorem 5 is very similar to our Theorem 2.18.…”

“…There thus appear to be significant links-or at least similarities-between the applications of the rich and currently quite distinct theories of graph limits and of hypergraph containers. The general "abstract" container results obtained in this paper and those obtained by Terry [58] may thus be seen as first steps towards an elucidation of those links.…”

The Method of Hypergraph Containers

“…In particular, Kühn, Osthus, Townsend, and Zhao [62] used it to describe the typical structure of oriented graphs without a transitive tournament of a given order. The recent independent works of Falgas-Ravry, O'Connell, Strömberg, and Uzzell [37] and of Terry [89] have developed a general framework for studying various enumeration problems in the setting of multicoloured graphs [37] and, more generally, in the very abstract setting of finite (model theoretic) structures [89]. In order to illustrate some of the ideas involved in applications of this kind, we will discuss the problem of counting finite metric spaces with bounded integral distances.…”