Abstract This article deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and was investigated by the first-named author. Here we study this class in Robinson’s logic, meaning the category of existentially closed models with embeddings following Haykazyan and Kirby, and prove that in this context this class is NSOP $_1$ and TP $_2$ .

Abstract The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

Abstract We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

Abstract We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th (ℤ , +, 1, 0, Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable, and supersimple. This is in contrast with Th (ℤ , +, 0, Pr, <) which is known to be undecidable by the works of Jockusch, Bateman, and Woods.

In this note we interpret Voevodsky's Univalence Axiom in the language of (abstract) model categories. We then show that any posetal locally Cartesian closed model category Qt in which the mapping Hom(w)(Z × B,C):Qt→Sets is functorial in Z and represented in Qt satisfies our homotopy version of the Univalence Axiom, albeit in a rather trivial way. This work was motivated by a question reported in [2], asking for a model of the Univalence Axiom not equivalent to the standard one.

Abstract In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss nonsplintering, an interesting notion that appears in the work of Rami Grossberg, Andrés Villaveces and Monica VanDieren, and we show that it is not trivial (in the sense that it can be different than splitting) whenever the directionality of the theory is not small. In the appendix we study dense types in RCF.

Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP2.