A unified approach to structural limits and limits of graphs with bounded tree-depth / Jaroslav Nešetřil, Patrice Ossona de Mendez.

Format:

Book

Author/Creator:

Nešetřil, Jaroslav, author.

Mendez, Patrice Ossona de, author.

Series:

Memoirs of the American Mathematical Society ; no. 1272.

Memoirs of the American Mathematical Society, 0065-9266 ; Number 1272

Language:

English

Subjects (All):

Model theory.

Functional analysis.

Trees (Graph theory).

Algebra, Boolean.

Graph theory.

Physical Description:

v, 108 pages : illustrations ; 24 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2020]

Summary:

In this paper the authors introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. The authors show how the various approaches to graph limits fit to this framework and that the authors naturally appear as "tractable cases" of a general theory. As an outcome of this, the authors provide extensions of known results. The authors believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, the authors consider limits of structures with bounded diameter connected components and prove that in this case the convergence can be "almost" studied component-wise. They also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, they consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role as "elementary bricks" these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of modeling the authors introduce here. Their example is also the first "intermediate class" with explicitly defined limit structures where the inverse problem has been solved.

Contents:

General theory

Modelings for sparse structures

Limits of graphs with bounded tree-depth.

Notes:

"January 2020, volume 263, number 1272 (second of 7 numbers)."

Includes bibliographical references (pages 105-108).

ISBN:

9781470440657

1470440652

OCLC:

1132241146