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First-Order Convergence and Roots†

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

Published online by Cambridge University Press:  24 February 2015

DEMETRES CHRISTOFIDES  and
DANIEL KRÁL’
DEMETRES CHRISTOFIDES
Affiliation:
School of Sciences, UCLan Cyprus, 12–14 University Avenue, 7080 Pyla, Cyprus (e-mail: dchristofides@uclan.ac.uk)
DANIEL KRÁL’
Affiliation:
Mathematics Institute, DIMAP and Department of Computer Science, University of Warwick, Coventry CV4 7AL, UK (e-mail: d.kral@warwick.ac.uk)
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Abstract

Nešetřil and Ossona de Mendez introduced the notion of first-order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether, if (Gi)i∈ℕ is a sequence of graphs with M being their first-order limit and v is a vertex of M, then there exists a sequence (vi)i∈ℕ of vertices such that the graphs Gi rooted at vi converge to M rooted at v. We show that this holds for almost all vertices v of M, and we give an example showing that the statement need not hold for all vertices.

MSC classification

Primary: 03C98: Applications of model theory
Secondary: 05C63: Infinite graphs 28E15: Other connections with logic and set theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 2 , March 2016 , pp. 213 - 221
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

†

This work was done during a visit to the Institut Mittag-Leffler (Djursholm, Sweden).

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