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 $\operatorname {acl}(B)$ divides over C. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an $\mathrm {NSOP}_1$ theory (...) with a complete type that forks, but does not divide, over a model (answering a question of d’Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$ -free incidence structures. (shrink)

We consider the question of when an expansion of a first-order topological structure has the property that every open set definable in the expansion is definable in the original structure. This question has been investigated by Dolich, Miller and Steinhorn in the setting of ordered structures as part of their work on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give (...) a further characterisation in the special case of an expansion by a generic predicate. (shrink)

We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

We continue the work on the relations between independence logic and the model-theoretic analysis of independence, generalizing the results of [15] and [16] to the framework of abstract independence relations for an arbitrary AEC. We give a model-theoretic interpretation of the independence atom and characterize under which conditions we can prove a completeness result with respect to the deductive system that axiomatizes independence in team semantics and statistics.

We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.

We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.

The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict independence (...) relation in the home sort, and hence is real rosy. In particular, if [Formula: see text] is o-minimal, then [Formula: see text] is real rosy. (shrink)

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)

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite-dimensional vector spaces over division rings.

We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”.

We introduce the notion of a w-good \-frame which is a weakening of Shelah’s notion of a good \-frame. Existence of a w-good \-frame implies existence of a model of size \. Tameness and amalgamation imply extension of a w-good \-frame to larger models. As an application we show:Theorem 0.1. Suppose\. If \ = \mathbb {I} = 1 \le \mathbb {I} < 2^{\lambda ^{++}}\)and\is\\)-tame, then\.The proof presented clarifies some of the details of the main theorem of Shelah and avoids using (...) the heavy set-theoretic machinery of Shelah by replacing it with tameness. (shrink)

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian (...) group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p -adic numbers is dp-minimal. (shrink)

We prove that if T is a complete theory with weak elimination of imaginaries, then there is an explicit bijection between strict independence relations for T and strict independence relations for Teq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T^{\rm eq}}$$\end{document}. We use this observation to show that if T is the theory of the Fraïssé limit of finite metric spaces with integer distances, then Teq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T^{\rm eq}}$$\end{document} has more than one (...) strict independence relation. This answers a question of Adler :1–20, 2009, Question 1.7). (shrink)

A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be (...) closely related to modular pairs in the lattice of algebraically closed sets. (shrink)

Let T be a complete geometric theory and let $$T_P$$ T P be the theory of dense pairs of models of T. We show that if T is superrosy with "Equation missing"-rank 1 then $$T_P$$ T P is superrosy with "Equation missing"-rank at most $$\omega $$ ω.

We study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p=\operatorname {tp}$ where $B=\operatorname {acl}$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$, the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure $\approx $ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total $\mathop {\smile \hskip -0.9em ^| \ (...) }^K$ -Morley sequence I in p, the weak canonical base of $\operatorname {tp}$ is $\operatorname {acl}$, if the hyperimaginary $I/\approx $ is eliminable to e, a sequence of imaginaries. We also supply a couple of criteria for when the weak canonical base of p exists. In particular the weak canonical base of p is the intersection of the weak canonical bases of all total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p over B. However, while we investigate some examples, we point out that given two weak canonical bases of total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p need not be interalgebraic, contrary to the case of simple theories. Lastly we suggest an independence relation relying on weak canonical bases, when T has those. The relation, satisfying transitivity and base monotonicity, might be useful in further studies on NSOP $_1$ theories. (shrink)

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some groundwork for extending these results to a power-bounded T-convex (...) theory. (shrink)

For n≥3, define Tn to be the theory of the generic Kn-free graph, where Kn is the complete graph on n vertices. We prove a graph-theoretic characterization of dividing in Tn and use it to show that forking and dividing are the same for complete types. We then give an example of a forking and nondividing formula. Altogether, Tn provides a counterexample to a question of Chernikov and Kaplan.

We develop the theory of domination by stable types and stable weight in an arbitrary theory.

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 study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over (...) a finite field. (shrink)