On VC-minimal theories and variants

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Archive for Mathematical Logic 52 (7-8):743-758 (2013)
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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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10.1007/s00153-013-0341-z

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Citations of this work

Semi-Equational Theories.Artem Chernikov & Alex Mennen - forthcoming - Journal of Symbolic Logic:1-32.
On VC-minimal fields and dp-smallness.Vincent Guingona - 2014 - Archive for Mathematical Logic 53 (5-6):503-517.
Hereditary G-compactness.Tomasz Rzepecki - 2021 - Archive for Mathematical Logic 60 (7):837-856.
On VC-Density in VC-Minimal Theories.Vincent Guingona - 2022 - Notre Dame Journal of Formal Logic 63 (3):395-413.

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References found in this work

Dp-Minimality: Basic Facts and Examples.Alfred Dolich, John Goodrick & David Lippel - 2011 - Notre Dame Journal of Formal Logic 52 (3):267-288.
Classification Theory and the Number of Nonisomorphic Models.S. Shelah - 1982 - Journal of Symbolic Logic 47 (3):694-696.
On uniform definability of types over finite sets.Vincent Guingona - 2012 - Journal of Symbolic Logic 77 (2):499-514.
Kueker's conjecture for stable theories.Ehud Hrushovski - 1989 - Journal of Symbolic Logic 54 (1):207-220.
Convexly orderable groups and valued fields.Joseph Flenner & Vincent Guingona - 2014 - Journal of Symbolic Logic 79 (1):154-170.

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