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Structure and enumeration theorems for hereditary properties in finite relational languages

Given a finite relational language $\calL$, a hereditary $\calL$-property is a class of finite $\calL$-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 $\calL$-properties, where $\calL$ 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 $\calL$-property. Given a hereditary $\calL$-property $\calH$, we prove an approximate asymptotic enumeration theorem for $\calH$ in terms of its generalized asymptotic density. Further we prove an approximate structure theorem for $\calH$, under the assumption of that $\calH$ has a stability theorem. The tools we use include a new application of the hypergraph containers theorem (Balogh-Morris-Samotij, Saxton-Thomason) to the setting of $\calL$-structures, a general supersaturation theorem for hereditary $\calL$-properties (also new), and a general graph removal lemma for $\calL$-structures proved by Aroskar and Cummings.