We study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are (...) indeed [Formula: see text]. (shrink)

This paper investigates a connection between the semantic notion provided by the ordering * among theories in model theory and the syntactic SOPn hierarchy of Shelah. It introduces two properties which are natural extensions of this hierarchy, called SOP2 and SOP1. It is shown here that SOP3 implies SOP2 implies SOP1. In Shelah's article 229) it was shown that SOP3 implies *-maximality and we prove here that *-maximality in a model of GCH implies a property called SOP2″. It has been (...) subsequently shown by Shelah and Usvyatsov that SOP2″ and SOP2 are equivalent, so obtaining an implication between *-maximality and SOP2. It is not known if SOP2 and SOP3 are equivalent. Together with the known results about the connection between the SOPn hierarchy and the existence of universal models in the absence of GCH, the paper provides a step toward the classification of unstable theories without the strict order property. (shrink)

Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T, canonical base of an amalgamation class P is the union of names of ψ-definitions of P, ψ ranging over stationary L-formulas in P. Also, we prove that the same is true with stable formulas for an 1-based theory having (...) elimination of hyperimaginaries. For such a theory, the stable forking property holds, too. (shrink)

We give definitions that distinguish between two notions of indiscernibility for a set {aη∣η∈ω>ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{a_{\eta} \mid \eta \in ^{\omega>}\omega\}}$$\end{document} that saw original use in Shelah [Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam, 1990], which we name s- and str−indiscernibility. Using these definitions and detailed proofs, we prove s- and str-modeling theorems and give applications of these theorems. In particular, we verify a step in the argument that TP is equivalent (...) to TP1 or TP2 that has not seen explication in the literature. In the Appendix, we exposit the proofs of Shelah [Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam, 1990, App. 2.6, 2.7], expanding on the details. (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)

Let A be an abelian variety in an algebraically closed field of characteristic 0. We prove that the expansion of A by a generic divisible subgroup of A with the same torsion exists provided A has few algebraic endomorphisms, namely. The resulting theory is NSOP1 and not simple. Note that there exist abelian varieties A with of any genus.