First order convergence of matroids

František Kardoš, Daniel Král’, Anita Liebenau, Lukáš Mach

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4 Citations (Scopus)

Abstract

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini–Schramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branch-depth assumption nor the representability assumption can be removed.

Original languageEnglish
Pages (from-to)150-168
Number of pages19
JournalEuropean Journal of Combinatorics
Volume59
DOIs
Publication statusPublished - 1 Jan 2017

Cite this

Kardoš, F., Král’, D., Liebenau, A., & Mach, L. (2017). First order convergence of matroids. European Journal of Combinatorics, 59, 150-168. https://doi.org/10.1016/j.ejc.2016.08.005