First order convergence of matroids. (January 2017)

• Record Type: Journal Article
• Authors: Kardoš, František; Král', Daniel; Liebenau, Anita; Mach, Lukáš
• Is Part Of: European journal of combinatorics. Volume 59(2017)
• Page Start: 150
• Page End: 168
• Publication Date: 2017-01
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2016.08.005
• Languages: English
• ISSNs: 0195-6698

Abstract

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.