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)

Disjoint n-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this article, we show that if a countably categorical theory T admits an expansion with disjoint n-amalgamation for all n, then T is pseudofinite. All theories which admit an expansion with disjoint n-amalgamation for all n are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain nonsimple theories are pseudofinite. As (...) case studies, we examine two generic theories of equivalence relations, Tfeq∗ and TCPZ, and show that both are pseudofinite. The theories Tfeq∗ and TCPZ are not simple, but they have NSOP1. This is established here for TCPZ for the first time. (shrink)

The author has previously shown that for a certain class of structures $\mathcal {I}$, $\mathcal {I}$-indexed indiscernible sets have the modeling property just in case the age of $\mathcal {I}$ is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. This result is applied to give new proofs that certain classes of trees are Ramsey. To aid this (...) project we develop the logic of EM-types. (shrink)

We introduce several concepts concerning the indiscernibility of trees. A tree is by definition an ordered set such that, for any a∈O, the initial segment {b∈O:b (...) A of the form η∈O, where O is a tree, and aη labeled by η is an element in M. Such a set A is also called a tree in this paper. We study the indiscernibility of trees A in general settings and apply the obtained results to the study of unstable theories. (shrink)

We prove the NTP1 property of a geometric theory T is inherited by theories of lovely pairs and H‐structures associated to T. We also provide a class of examples of nonsimple geometric NTP1 theories.

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)

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$. This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$, such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$, such as Kim-independence or forking independence, and describe interactions between them.

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.