Abstract Let $\mathcal M=(M,<,\ldots)$ be a linearly ordered first-order structure and T its complete theory. We investigate conditions for T that could guarantee that $\mathcal M$ is not much more complex than some colored orders (linear orders with added unary predicates). Motivated by Rubin’s work [5], we label three conditions expressing properties of types of T and/or automorphisms of models of T. We prove several results which indicate the “geometric” simplicity of definable sets in models of theories satisfying these conditions. For example, we prove that the strongest condition characterizes, up to definitional equivalence (inter-definability), theories of colored orders expanded by equivalence relations with convex classes.

Abstract We study nonorthogonality of symmetric, regular types and show that it preserves generic stability and is an equivalence relation on the set of all generically stable, regular types. We prove that some of the nice properties from the stable context hold in general. In the case of strongly regular types we will relate to the global Rudin–Keisler order.

Abstract We prove that a Kueker theory with infinite dcl(∅) does not have the strict order property and that strongly minimal types are dense: any non-algebraic formula is contained in a strongly minimal type.

Abstract We answer a question of Cassidy and Kolchin about the universality of the constrained closure of a differential field by working in a larger category of models.

Pillay studied nonmultidimensional superstable theories in [8], among other things defining a certain hierarchy of regular types in terms of which all other types may be analysed. Using this hierarchy, he showed that after naming a suitable ‘base’ of parameters, there are j-constructible (hence locally atomic) models over arbitrary sets (see Section 2 for definitions). It is asked at the end of [8] whether the parameter set can be removed. On a different note, it has been known for some time that in nonmultidimensional superstable theories, R∞-rank is definable for formulas having finite rank (see for example [9]). Definability of R∞-rank has had various applications in the literature, and so it is natural to ask whether the restriction to finite rank is necessary. In this paper we do not quite answer this question, but instead use Pillay's analysis to establish the existence of a ‘new’ continuous rank (the original idea for which is due to Tanović) which is defined on all complete types, reflects forking as does R∞-rank and satisfies certain definability properties.

Abstract We prove: Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding has 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.