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On VC-minimal theories and variants

Abstract

In this paper, we study VC-minimal theories and explore related concepts. We first define the notion of convex orderablity and show that this lies strictly between VC-minimality and dp-minimality. To do this we prove a general result about set systems with independence dimension ≤ 1. Next, we define the notion of weak VC-minimality, show it lies strictly between VC-minimality and dependence, and show that all unstable weakly VC-minimal theories interpret an infinite linear order. Finally, we define the notion full VC-minimality, show that this lies strictly between weak o-minimality and VC-minimality, and show that theories that are fully VC-minimal have low VC-density.

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Correspondence to Vincent Guingona.

Additional information

Vincent Guingona and Michael C. Laskowski were partially supported by Laskowski’s NSF Grants DMS-0600217 and 0901336.

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Guingona, V., Laskowski, M.C. On VC-minimal theories and variants. Arch. Math. Logic 52, 743–758 (2013). https://doi.org/10.1007/s00153-013-0341-z

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  • DOI: https://doi.org/10.1007/s00153-013-0341-z

Keywords

  • VC-minimal
  • UDTFS
  • Convexly orderable
  • NIP
  • Dependent
  • Kueker conjecture

Mathematics Subject Classification (2010)

  • Primary 03C45