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 forking assuming NTP 2 .

We study NSOP _1 theories. We define Kim-independence , which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim’s lemma, local character, symmetry, and an independence theorem, and that moreover these properties individually characterize NSOP _1 theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.

We show that NIP fields have no Artin-Schreier extension, and that simple fields have only a finite number of them.

Abstract. We show that if the restriction of the Lascar equivalence relation to a KP-strong type is non-trivial, then it is non-smooth (when viewed as a Borel equivalence relation on an appropriate space of types).

The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density. introductionThis paper grew out of discussions that the authors had during a meeting in Oberwolfach in January 2010, following a talk of Deirdre Haskell, and conversations with Sergei Starchenko, on their recent joint work with Aschenbrenner, Dolich and Macpherson [2]. Haskell's talk made it apparent to us that the notion of VC-density (Vapnik-Chervonenkis density), investigated in [2], is closely related to "dependence rank" (dp-rank) introduced by the third author in [14]. Discussions with Starchenko helped us realize that certain questions, such as additivity, which were (and still are, to our knowledge) open for VC-density, may be approached more easily in the context of dp-rank. This paper is the first step in the program of investigating basic properties of dp-rank and its connections with VC-density.Whereas dp-rank is a relatively new notion, VC-density and related concepts have been studied for quite some time in the frameworks of machine learning, computational geometry, and other branches of theoretical computer science. Recent developments point to a connection between VC-density and dp-rank, strengthening the bridge between model theory and these subjects. We believe that investigating properties of dp-rank is important for discovering the nature of this connection. Furthermore, once this relation is better understood, theorems about dp-rank are likely to prove useful in the study of finite and infinite combinatorics related to VC-classes.Dp-rank was originally defined in [14] as an attempt to capture how far a certain type (or a theory) is from having the independence property. It also helped us to isolate a minimality notion of dependence for types and theories (that is, having rank 1). We called this notion dp-minimality and investigated it in [7]. Both dprank and dp-minimality were simplifications of Shelah's various ranks from [12],