  In this paper, we study VC-minimal theories and explore related concepts. We
first define the notion of convex orderablility and show that this lies
strictly between VC-minimality and dp-minimality. 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.

Logic