A first order theory T of power λ is called unidimensional if any twoλ+-saturated models of T of the same cardinality are isomorphic. We prove here that such theories are superstable, solving a problem of Shelah. The proof involves an existence theorem and a definability theorem for definable groups in stable theories, and an analysis of their relation to regular types.

T is stable. We define the notion of meager regular type and prove that a meager regular type is locally modular. Assuming I < 2o and G is a definable abelian group with locally modular regular generics, we prove a counterpart of Saffe's conjecture. Using these results, for superstable T we prove the conjecture of vanishing multiplicities. Also, as a further application, in some additional cases we prove a conjecture regarding topological stability of pseudo-types over Q.

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. (shrink)

We prove here:Theorem. LetT be a countable complete superstable non ω-stable theory with fewer than continuum many countable models. Then there is a definable groupG with locally modular regular generics, such thatG is not connected-by-finite and any type inG eq orthogonal to the generics has Morley rank.Corollary. LetT be a countable complete superstable theory in which no infinite group is definable. ThenT has either at most countably many, or exactly continuum many countable models, up to isomorphism.