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Structural convergence and algebraic roots

Part of: Miscellaneous topics in measure theory Graph theory Model theory

Published online by Cambridge University Press:  23 December 2024

David Hartman Open the ORCID record for David Hartman [Opens in a new window] ,
Tomáš Hons Open the ORCID record for Tomáš Hons [Opens in a new window]  and
Jaroslav Nešetřil Open the ORCID record for Jaroslav Nešetřil [Opens in a new window]
David Hartman*
Affiliation:
Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic Institute of Computer Science of the Czech Academy of Sciences, Prague, Czech Republic
Tomáš Hons
Affiliation:
Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Jaroslav Nešetřil
Affiliation:
Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
*
Corresponding author: David Hartman; Email: hartman@iuuk.mff.cuni.cz
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Abstract

Structural convergence is a framework for the 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 $(G_n)$ converging to a limit $L$ and a vertex $r$ of $L$, it is possible to find a sequence of vertices $(r_n)$ such that $L$ rooted at $r$ is the limit of the graphs $G_n$ rooted at $r_n$. 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 on 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 $(r_n)$ always exists.

Keywords

Structural convergencegraph limitsrooting verticesalgebraic sets

MSC classification

Primary: 03C98: Applications of model theory 05C63: Infinite graphs
Secondary: 28E15: Other connections with logic and set theory

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 34 , Issue 3 , May 2025 , pp. 392 - 400
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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