Let L be a finite relational language and α=(αR:R ∈ L) a tuple with 0 < αR ≤1 for each R ∈ L. Consider a dimension function $ \delta _\alpha (A) = \left| A \right| - \sum\limits_{R \in L} {\alpha {\mathop{\rm Re}\nolimits} R(A)} $ where each eR(A) is the number of realizations of R in A. Let $K_\alpha $ be the class of finite structures A such that $\delta _\alpha (X) \ge 0$ 0 for any substructure X of A. We (...) show that the theory of the generic model of $K_\alpha $ is AE-axiomatizable for any α. (shrink)

A theory T is called almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical if for any pure types p1,…,pn there are only finitely many pure types which extend p1 ∪…∪pn. It is shown that if T is an almost [MATHEMATICAL SCRIPT CAPITAL N]0-categorical theory with I = 3, then a dense linear ordering is interpretable in T.

An infinite structure is said to be minimal if each of its definable subset is finite or cofinite. Modifying Hrushovski's method we construct minimal, non strongly minimal structures with arbitrary finite dimensions. This answers negatively to a problem posed by B. I Zilber.

Let L be a countable relational language. Baldwin asked whether there is an ab initio generic L-structure which is superstable but not ω-stable. We give a positive answer to his question, and prove that there is no ab initio generic L-structure which is superstable but not ω-stable, if L is finite and the generic is saturated.

We construct an ab initio generic structure for a predimension function with a positive rational coefficient less than or equal to 1 which is unsaturated and has a superstable non-ω-stable theory.

We show that if a class K of finite relational structures is closed under quasi-substructures, then there is no saturated K-generic structure that is superstable but not ω -stable.

A partial map f of a structure M is called almost total if |M — dom| = |M — ran| < ω. We study a difference between an almost total elementary map and an automorphism.

Hrushovski constructed an -categorical stable pseudoplane which refuted Lachlan's conjecture. In this note, we show that an -categorical projective plane cannot be constructed by "the Hrushovski method.".