We prove: Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding has 2 ℵ 0 nonisomorphic countable models. Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.

In this paper we establish the following theorems. THEOREM A. Let T be a complete first-order theory which is uncountable. Then: (i) I(|T|, T) ≥ ℵ 0 . (ii) If T is not unidimensional, then for any λ ≥ |T|, I (λ, T) ≥ ℵ 0 . THEOREM B. Let T be superstable, not totally transcendental and nonmultidimensional. Let θ(x) be a formula of least R ∞ rank which does not have Morley rank, and let p be any stationary completion (...) of θ which also fails to have Morley rank. Then p is regular and locally modular. (shrink)

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.

We try to count the number of countable models M of T with a fixed set Q = φ (M) of realizations of a type φ. Also, for stable T, we define an ordinal rank M measuring multiplicity of types, with additivity properties similar to those of U-rank.

Let T be an uncountable, superstable theory. In this paper we prove Theorem A. If T has finite rank, then I(|T|, T) ≥ ℵ0. Theorem B. If T is trivial, then I(|T|, T) ≥ ℵ0.

We construct a new class of 1 categorical structures, disproving Zilber's conjecture, and study some of their properties.