A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F20%3A10422246" target="_blank" >RIV/00216208:11320/20:10422246 - isvavai.cz</a>
Výsledek na webu
<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=Q~AWcwuUkW" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=Q~AWcwuUkW</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1090/memo/1272" target="_blank" >10.1090/memo/1272</a>
Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
Popis výsledku v původním jazyce
In this paper we 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. We show how the various approaches to graph limits fit to this framework and that they naturally appear as "tractable cases" of a general theory. As an outcome of this, we provide extensions of known results. We believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be "almost" studied component-wise. We also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, we 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 we introduce here. Our example is also the first "intermediate class" with explicitly defined limit structures where the inverse problem has been solved.
Název v anglickém jazyce
A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
Popis výsledku anglicky
In this paper we 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. We show how the various approaches to graph limits fit to this framework and that they naturally appear as "tractable cases" of a general theory. As an outcome of this, we provide extensions of known results. We believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be "almost" studied component-wise. We also propose the structure of limit objects for convergent sequences of sparse structures. Eventually, we 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 we introduce here. Our example is also the first "intermediate class" with explicitly defined limit structures where the inverse problem has been solved.
Klasifikace
Druh
J<sub>imp</sub> - Článek v periodiku v databázi Web of Science
CEP obor
—
OECD FORD obor
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Návaznosti výsledku
Projekt
—
Návaznosti
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Ostatní
Rok uplatnění
2020
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název periodika
Memoirs of the American Mathematical Society
ISSN
0065-9266
e-ISSN
—
Svazek periodika
263
Číslo periodika v rámci svazku
1272
Stát vydavatele periodika
US - Spojené státy americké
Počet stran výsledku
109
Strana od-do
1-109
Kód UT WoS článku
000516766100001
EID výsledku v databázi Scopus
2-s2.0-85080910455