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 line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.

We consider existentially closed fields with several orderings, valuations, and [Formula: see text]-valuations. We show that these structures are NTP2 of finite burden, but usually have the independence property. Moreover, forking agrees with dividing, and forking can be characterized in terms of forking in ACVF, RCF, and [Formula: see text]CF.

We consider existentially closed fields with several orderings, valuations, and p-valuations. We show that these structures are NTP2 of finite burden, but usually have the independence property. Mo...

Motivated by the Ax–Kochen/Ershov principle, a large number of questions about Henselian valued fields have been shown to reduce to analogous questions about the value group and residue field. In this article, we investigate the burden of Henselian valued fields in the three-sorted Denef–Pas language. If T is a theory of Henselian valued fields admitting relative quantifier elimination (in any characteristic), we show that the burden of T is equal to the sum of the burdens of its value group and (...) residue field. As a consequence, T is NTP2 if and only if its residue field and value group are; the same is true for the statements “T is strong” and “T has finite burden.”. (shrink)

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 ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any (...) theory is always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic is NTP2 if and only if the residue field is NTP2 , so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2. (shrink)

We show that if T is any geometric theory having the NTP2 then the corresponding theories of lovely pairs of models of T and of H‐structures associated to T also have the NTP2. We also prove that if T is strong then the same two expansions of T are also strong.

We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T. We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP2. In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP2 exactly when has these properties. We (...) show that if T is superrosy of thorn rank 1, then so is, and that the converse holds if T satisfies acl = dcl. (shrink)

Journal of Mathematical Logic, Ahead of Print. Given a structure [math] and a stably embedded [math]-definable set [math], we prove tameness preservation results when enriching the induced structure on [math] by some further structure [math]. In particular, we show that if [math] and [math] are stable (respectively, superstable, [math]-stable), then so is the theory [math] of the enrichment of [math] by [math]. Assuming simplicity of [math], elimination of hyperimaginaries and a further condition on [math] related to the behavior of algebraic (...) closure, we also show that simplicity and NSOP1 pass from [math] to [math]. We then prove several applications for tame expansions of weakly minimal structures and, in particular, the group of integers. For example, we construct the first known examples of strictly stable expansions of [math]. More generally, we show that any stable (respectively, superstable, simple, NIP, NTP2, NSOP1) countable graph can be defined in a stable (respectively, superstable, simple, NIP, NTP2, NSOP1) expansion of [math] by some unary predicate [math]. (shrink)

Given a model-complete theory of topological fields, we considered its generic differential expansions and under a certain hypothesis of largeness, we axiomatised the class of existentially closed ones. Here we show that a density result for definable types over definably closed subsets in such differential topological fields. Then we show two transfer results, one on the VC-density and the other one, on the combinatorial property NTP2.

We prove the preservation theorems for NATP; many of them extend the previously established preservation results for other model-theoretic tree properties. Using them, we also furnish proper examples of NATP theories which are simultaneously TP2 and SOP. First, we show that NATP is preserved by the parametrization and sum of the theories of Fraïssé limits of Fraïssé classes satisfying strong amalgamation property. Second, the preservation of NATP for two kinds of dense/co-dense expansions, i.e. the theories of lovely pairs and of (...) [Formula: see text]-structures for geometric theories, and dense/co-dense expansion on vector spaces is proved. Next, we prove the preservation of NATP for the generic predicate expansion and the pair of an algebraically closed field and its distinguished subfield; for the latter, not only NATP, but also the preservation of NTP1 and NTP2 is considered. Finally, we present some proper examples of NATP theories using the results proved in this paper, including the parametrization of DLO and the expansion of an algebraically closed field by adding a generic linear order. In particular, we show that the model companion of the theory of algebraically closed fields with circular orders (ACFO) is NATP. (shrink)

We introduce the concept of o-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. Along with several examples and non-examples of these classes, we present some classification theory results of their infinite ultraproducts: Every infinite ultraproduct of structures in an o-asymptotic class is superrosy of U^þ-rank 1, and NTP2 (in fact, inp-minimal).

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 symmetric. Based on this study, we develop notions of weight which characterize NTP2, dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which (...) are witnesses. As a by-product we obtain information on types co-dominated by generically stable types in dependent theories. For example, we prove that every Morley sequence in such a type is a witness. (shrink)

As a continuation of ideas initiated in [19], we study bi-colored (generic) expansions of geometric theories in the style of the Fraïssé–Hrushovski construction method. Here we examine that the properties $NTP_{2}$, strongness, $NSOP_{1}$, and simplicity can be transferred to the expansions. As a consequence, while the corresponding bi-colored expansion of a red non-principal ultraproduct of p-adic fields is $NTP_{2}$, the expansion of algebraically closed fields with generic automorphism is a simple theory. Furthermore, these theories are strong with $\operatorname {\mathrm {bdn}}(\text (...) {"}x=x\text {"})=(\aleph _0)_{-}$. (shrink)