Related papers: An independence theorem for NTP2 theories

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…

We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular it follows from our investigation that in resilient theories strict non-forking is…

Forking is a central notion of model theory, generalizing linear independence in vector spaces and algebraic independence in fields. We develop the theory of forking in abstract, category-theoretic terms, for reasons both practical (we…

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP$_{1}$ theory. We deduce symmetry of Kim-independence and the independence…

We prove some results about the theory of independence in $\mathrm{NSOP}_{3}$ theories that do not hold in $\mathrm{NSOP}_{4}$ theories. We generalize Chernikov's work on simple and co-simple types in $\mathrm{NTP}_{2}$ theories to types…

Kim's Lemma is a key ingredient in the theory of forking independence in simple theories. It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters. Variants of Kim's Lemma have formed the…

Monadic stability and the more general monadic dependence (or NIP) are tameness conditions for classes of logical structures, studied in the 80's in Shelah's classification program in model theory. They recently emerged in algorithmic and…

We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a…

We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2 then there…

Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion…

An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is…

The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one…

We introduce a family of local ranks DQ depending on a finite set Q of pairs of the form (\varphi(x,y),q(y)) where \varphi(x,y) is a formula and q(y) is a global type. We prove that in any NSOP1 theory these ranks satisfy some desirable…

We try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory…

We adapt the properties of Kim-independence in NSOP1 theories with existence proven in [5],[4] and [2] by Ramsey, Kaplan, Chernikov, Dobrowolski and Kim to hyperimaginaries by adding the assumption of existence for hyperimaginaries. We show…

We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically closed subsets,…

We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type $p$ over a set $B$ does not divide over $C\subseteq B$, then no extension of $p$ to a complete type over $\text{acl}(B)$ divides over $C$.…

We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of the properties needed for an adequate geometric…

An important dividing line in the class of unstable theories is being NSOP$_1$, which is more general than being simple. In NSOP$_1$ theories forking independence may not be as well-behaved as in stable or simple theories, so it is replaced…

We consider existentially closed fields with several orderings, valuations, and $p$-valuations. We show that these structures are NTP$_2$ of finite burden, but usually have the independence property. Moreover, forking agrees with dividing,…