Nešetřil, Jaroslav; Ossona de Mendez, Patrice A unified approach to structural limits and limits of graphs with bounded tree-depth. (English) Zbl 1491.03004 Memoirs of the American Mathematical Society 1272. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4065-7/pbk; 978-1-4704-5652-8/ebook). v, 108 p. (2020). This monograph sets up a framework for studying limits of infinite sequences of finite relational structures, such as graphs, by associating, when possible, a “limit object”, called modeling, to such a sequence. This framework generalizes the frameworks considered by graph theorists when studying the notions of graphon and graphing. To say something about the main results we need to know something about three central concepts of this framework.A sequence of finite structures \((A_n)_{n \in \mathbb{N}}\) (having the same finite relational signature) is FO-convergent if for every first-order formula \(\varphi(x_1, \ldots, x_k)\), the proportion of \(k\)-tuples in \(A_n\) which satisfy \(\varphi\) converges as \(n\) tends to infinity.A modeling is a relational structure with the additional features that its domain is a Borel space equipped with a probability measure and every first-order definable relation is measurable in the corresponding product \(\sigma\)-algebra.A modeling \(M\) is called a modeling FO-limit of an FO-convergent sequence of finite structures \((A_n)_{n \in \mathbb{N}}\) if for every first-order formula the proportion of tuples in \(A_n\) which satisfy it converges to the probability (according to \(M\)) of the relation defined by that formula in \(M\).The main definitions and results are collected in the first section of the monograph. The first of these states that if \(\mathcal{C}\) is a monotone class of finite graphs such that every \(FO\)-convergent sequence of graphs from \(\mathcal{C}\) has a modeling FO-limit, then \(\mathcal{C}\) is nowhere dense. It follows that there are FO-convergent sequences of graphs without a modeling FO-limit. It is conjectured that the converse implication also holds and Section 5.2 in the appendix reports on recent progress towards verifying the conjecture.The remaining main results are concerned with (colored) graphs: (a) Every FO-convergent sequence of finite graphs with a fixed maximum degree has a modeling FO-limit, and (b) every FO-convergent sequence of finite colored trees with a fixed maximum height has a modeling FO-limit. The statement (b) is generalized (see Theorem 4.3.6), via the use of interpretations, to say roughly that (c) every FO-convergent sequence of finite colored graphs with fixed maximum tree height has a modeling FO-limit. Conjecture 1.3 proposes conditions under which a modeling should be a modeling FO-limit of an FO-convergent sequence of finite graphs of with fixed maximum degree. In the case of the statements (b) and (c), this monograph proves converses of the implications. More precisely: If \(M\) is a modeling such that its underlying first-order structure is a rooted colored tree with finite height and if \(M\) satisfies the Finitary Mass Transport Principle (FMTP) (Definition 4.20), then \(M\) is a modeling FO-limit of a sequence of finite rooted colored trees. Via the use of interpretations this result, which is technically hard to prove, can be generalized (in Theorem 4.3.6) to roughly the following: If \(M\) is a modeling that can be interpreted in a modeling \(T\) with the FMTP such that the underlying first-order structure of \(T\) is a rooted colored tree, then \(M\) is the modeling FO-limit of a sequence of finite graphs with a fixed bound on the tree height.Section 2 develops general methods (for possible future use) involving (Lindenbaum-Tarski) Boolean algebras, Stone spaces, Ehrenfeucht-Fraisse games, the Gaifmain locality theorem and the notion of an interpretation of one structure in another. Section 3 studies modelings and their relationship to the concepts of Vapnik-Chervonenkis dimension, nowhere denseness, and random freeness. Section 4 studies sequences and limits of (colored) graphs with bounded tree-height, in particular rooted colored trees with bounded height, and the results mentioned above about such structures are proved here. The final Section 5 discusses open problems related to the theory developed in the monograph. Reviewer: Vera Koponen (Uppsala) Cited in 6 Documents MSC: 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03C13 Model theory of finite structures 03C98 Applications of model theory 05C99 Graph theory 06E15 Stone spaces (Boolean spaces) and related structures 28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures Keywords:graph and relational structure; graph limits; structural limits; Radon measures; Stone space; model theory; first-order logic; measurable graph PDFBibTeX XMLCite \textit{J. Nešetřil} and \textit{P. 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