Softcover ISBN:  9781470440657 
Product Code:  MEMO/263/1272 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
eBook ISBN:  9781470456528 
Product Code:  MEMO/263/1272.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
Softcover ISBN:  9781470440657 
eBook: ISBN:  9781470456528 
Product Code:  MEMO/263/1272.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $102.00 $76.50 
Softcover ISBN:  9781470440657 
Product Code:  MEMO/263/1272 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
eBook ISBN:  9781470456528 
Product Code:  MEMO/263/1272.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
Softcover ISBN:  9781470440657 
eBook ISBN:  9781470456528 
Product Code:  MEMO/263/1272.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $102.00 $76.50 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 263; 2020; 108 ppMSC: Primary 03; 05; 06; Secondary 28
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 componentwise. 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 treedepth, 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 firstorder 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.

Table of Contents

Chapters

1. Introduction

2. General Theory

3. Modelings for Sparse Structures

4. Limits of Graphs with Bounded Treedepth

5. Concluding Remarks


Additional Material

RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
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 componentwise. 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 treedepth, 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 firstorder 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.

Chapters

1. Introduction

2. General Theory

3. Modelings for Sparse Structures

4. Limits of Graphs with Bounded Treedepth

5. Concluding Remarks