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arXiv · 1501.06518

First order convergence of matroids

Frantisek Kardos·Daniel Kral·Anita Liebenau·Lukas Mach
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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.

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Frantisek Kardos, Daniel Kral, Anita Liebenau, Lukas Mach. 2016-08-13. First order convergence of matroids. https://arxiv.org/abs/1501.06518

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