Rooting algebraic vertices of convergent sequences
Result description
Structural convergence is a framework for convergence of graphs by Nešetřil and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs (Gn) converging to a limit L and a vertex r of L it is possible to find a sequence of vertices (rn) such that L rooted at r is the limit of the graphs Gn rooted at rn. A counterexample was found by Christofides and Král’, but they showed that the statement holds for almost all vertices r of L. We offer another perspective to the original problem by considering the size of definable sets to which the root r belongs. We prove that if r is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots (rn) always exists.
Keywords
The result's identifiers
Result code in IS VaVaI
Result on the web
https://journals.phil.muni.cz/eurocomb/article/view/35609/31523
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Rooting algebraic vertices of convergent sequences
Original language description
Structural convergence is a framework for convergence of graphs by Nešetřil and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs (Gn) converging to a limit L and a vertex r of L it is possible to find a sequence of vertices (rn) such that L rooted at r is the limit of the graphs Gn rooted at rn. A counterexample was found by Christofides and Král’, but they showed that the statement holds for almost all vertices r of L. We offer another perspective to the original problem by considering the size of definable sets to which the root r belongs. We prove that if r is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots (rn) always exists.
Czech name
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Czech description
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Classification
Type
D - Article in proceedings
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Article name in the collection
EUROCOMB’23. Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications
ISBN
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ISSN
2788-3116
e-ISSN
2788-3116
Number of pages
6
Pages from-to
539-544
Publisher name
MUNI Press
Place of publication
Brno
Event location
Prague
Event date
Aug 28, 2023
Type of event by nationality
EUR - Evropská akce
UT code for WoS article
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Basic information
Result type
D - Article in proceedings
OECD FORD
Pure mathematics
Year of implementation
2023