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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

rootingalgebraic verticesconvergent sequences

The result's identifiers

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

  • Czech description

Classification

  • Type

    D - Article in proceedings

  • CEP classification

  • OECD FORD branch

    10101 - Pure mathematics

Result continuities

  • Project

  • 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

  • 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

Basic information

Result type

D - Article in proceedings

D

OECD FORD

Pure mathematics

Year of implementation

2023