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  1. On Properties of Theories Which Preclude the Existence of Universal Models.Mirna Džamonja & Saharon Shelah - 2006 - Annals of Pure and Applied Logic 139 (1):280-302.
    We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality λ when certain cardinal arithmetic assumptions about λ implying the failure of GCH hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to (...)
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  • Club Guessing and the Universal Models.Mirna Džamonja - 2005 - Notre Dame Journal of Formal Logic 46 (3):283-300.
    We survey the use of club guessing and other PCF constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal, element.
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  • A Remark on Strict Independence Relations.Gabriel Conant - 2016 - Archive for Mathematical Logic 55 (3-4):535-544.
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  • Forking and Dividing in Henson Graphs.Gabriel Conant - 2017 - Notre Dame Journal of Formal Logic 58 (4):555-566.
    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.
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  • Neostability in Countable Homogeneous Metric Spaces.Gabriel Conant - 2017 - Annals of Pure and Applied Logic 168 (7):1442-1471.
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  • Characterizing Model-Theoretic Dividing Lines Via Collapse of Generalized Indiscernibles.Vincent Guingona, Cameron Donnay Hill & Lynn Scow - 2017 - Annals of Pure and Applied Logic 168 (5):1091-1111.
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  • The Characteristic Sequence of a First-Order Formula.M. E. Malliaris - 2010 - Journal of Symbolic Logic 75 (4):1415-1440.
    For a first-order formula φ(x; y) we introduce and study the characteristic sequence ⟨P n : n < ω⟩ of hypergraphs defined by P n (y₁…., y n ):= $(\exists x)\bigwedge _{i\leq n}\varphi (x;y_{i})$ . We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of φ and vice versa. The main results are a characterization of NIP and of simplicity in terms of persistence of configurations in the characteristic sequence. Specifically, we show that (...)
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  • A Geometric Introduction to Forking and Thorn-Forking.Hans Adler - 2009 - Journal of Mathematical Logic 9 (1):1-20.
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  • Constructing the Hyperdefinable Group From the Group Configuration.Tristram de Piro, Byunghan Kim & Jessica Millar - 2006 - Journal of Mathematical Logic 6 (2):121-139.
  • Independence in Generic Incidence Structures.Gabriel Conant & Alex Kruckman - 2019 - Journal of Symbolic Logic 84 (2):750-780.
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  • 2003 Annual Meeting of the Association for Symbolic Logic.Andreas Blass - 2004 - Bulletin of Symbolic Logic 10 (1):120-145.
  • More on SOP 1 and SOP 2.Saharon Shelah & Alexander Usvyatsov - 2008 - Annals of Pure and Applied Logic 155 (1):16-31.
    This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 229–255] and [M. Džamonja, S. Shelah, On left triangle, open*-maximality, Annals of Pure and Applied Logic 125 119–158]. We present a rank function for NSOP1 theories and give an example of a theory which is NSOP1 but not simple. We also investigate the connection between maximality in the ordering left triangle, open* among complete first order theories and the SOP2 property. We (...)
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  • Some Remarks on Generic Structures.David M. Evans & Mark Wing Ho Wong - 2009 - Journal of Symbolic Logic 74 (4):1143 - 1154.
    We show that the N₀-categorical structures produced by Hrushovski's predimension construction with a control function fit neatly into Shelah's $SOP_n $ hierarchy: if they are not simple, then they have SOP₃ and NSOP₄. We also show that structures produced without using a control function can be undecidable and have SOP.
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  • Forking and Independence in o-Minimal Theories.Alfred Dolich - 2004 - Journal of Symbolic Logic 69 (1):215-240.
  • Universal Graphs at the Successor of a Singular Cardinal.Mirna Džamonja & Saharon Shelah - 2003 - Journal of Symbolic Logic 68 (2):366-388.
    The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality ℵ0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for some such μ there are $\mu^{++}$ graphs on μ+ that taken jointly are universal for the graphs on μ+, while $2^{\mu^+} \gg \mu^{++}$ . The paper also addresses the general problem of obtaining a framework for consistency results at the (...)
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  • Independence Property and Hyperbolic Groups.Eric Jaligot, Alexey Muranov & Azadeh Neman - 2008 - Bulletin of Symbolic Logic 14 (1):88 - 98.
    In continuation of [JOH04, OH07], we prove that existentially closed CSA-groups have the independence property. This is done by showing that there exist words having the independence property relative to the class of torsion-free hyperbolic groups.
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  • Hypergraph Sequences as a Tool for Saturation of Ultrapowers.M. E. Malliaris - 2012 - Journal of Symbolic Logic 77 (1):195-223.
    Let T 1 , T 2 be countable first-order theories, M i ⊨ T i , and any regular ultrafilter on λ ≥ $\aleph_{0}$ . A longstanding open problem of Keisler asks when T 2 is more complex than T 1 , as measured by the fact that for any such λ, , if the ultrapower (M 2 ) λ / realizes all types over sets of size ≤ λ, then so must the ultrapower (M 1 ) λ /. (...)
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  • Completeness and Categoricity : Formalization Without Foundationalism.John T. Baldwin - 2014 - Bulletin of Symbolic Logic 20 (1):39-79.
  • On ◁∗-Maximality.Mirna Džamonja & Saharon Shelah - 2004 - Annals of Pure and Applied Logic 125 (1-3):119-158.
    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 (...)
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  • On the Existence of Universal Models.Mirna Džamonja & Saharon Shelah - 2004 - Archive for Mathematical Logic 43 (7):901-936.
    Suppose that λ=λ <λ ≥ℵ0, and we are considering a theory T. We give a criterion on T which is sufficient for the consistent existence of λ++ universal models of T of size λ+ for models of T of size ≤λ+, and is meaningful when 2λ +>λ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind (...)
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  • Independence, Order, and the Interaction of Ultrafilters and Theories.M. E. Malliaris - 2012 - Annals of Pure and Applied Logic 163 (11):1580-1595.
    We consider the question, of longstanding interest, of realizing types in regular ultrapowers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a single formula. In this article, we analyze a class of formulas φ whose associated characteristic sequence of hypergraphs can be seen as describing realization of first- and second-order types in ultrapowers on one hand, and (...)
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  • Theories Without the Tree Property of the Second Kind.Artem Chernikov - 2014 - Annals of Pure and Applied Logic 165 (2):695-723.
    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 (...)
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  • Notions Around Tree Property 1.Byunghan Kim & Hyeung-Joon Kim - 2011 - Annals of Pure and Applied Logic 162 (9):698-709.
    In this paper, we study the notions related to tree property 1 , or, equivalently, SOP2. Among others, we supply a type-counting criterion for TP1 and show the equivalence of TP1 and k- TP1. Then we introduce the notions of weak k- TP1 for k≥2, and also supply type-counting criteria for those. We do not know whether weak k- TP1 implies TP1, but at least we prove that each weak k- TP1 implies SOP1. Our generalization of the tree-indiscernibility results in (...)
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  • Realization of Φ -Types and Keisler’s Order.M. E. Malliaris - 2009 - Annals of Pure and Applied Logic 157 (2-3):220-224.
    We show that the analysis of Keisler’s order can be localized to the study of φ-types. Specifically, if is a regular ultrafilter on λ such that and M is a model whose theory is countable, then is λ+-saturated iff it realizes all φ-types of size λ.
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  • Edge Distribution and Density in the Characteristic Sequence.M. E. Malliaris - 2010 - Annals of Pure and Applied Logic 162 (1):1-19.
    The characteristic sequence of hypergraphs Pn:n<ω associated to a formula φ, introduced in Malliaris [5], is defined by Pn=i≤nφ. We continue the study of characteristic sequences, showing that graph-theoretic techniques, notably Szemerédi’s celebrated regularity lemma, can be naturally applied to the study of model-theoretic complexity via the characteristic sequence. Specifically, we relate classification-theoretic properties of φ and of the Pn to density between components in Szemerédi-regular decompositions of graphs in the characteristic sequence. In addition, we use Szemerédi regularity to calibrate (...)
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  • More on SOP1 and SOP2.Saharon Shelah & Alexander Usvyatsov - 2008 - Annals of Pure and Applied Logic 155 (1):16-31.
    This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 229–255] and [M. Džamonja, S. Shelah, On left triangle, open*-maximality, Annals of Pure and Applied Logic 125 119–158]. We present a rank function for NSOP1 theories and give an example of a theory which is NSOP1 but not simple. We also investigate the connection between maximality in the ordering left triangle, open* among complete first order theories and the SOP2 property. We (...)
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  • Model Theoretic Properties of the Urysohn Sphere.Gabriel Conant & Caroline Terry - 2016 - Annals of Pure and Applied Logic 167 (1):49-72.
  • An Axiomatic Approach to Free Amalgamation.Gabriel Conant - 2017 - Journal of Symbolic Logic 82 (2):648-671.
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  • Model-Theoretic Properties of Ultrafilters Built by Independent Families of Functions.M. Malliaris & S. Shelah - 2014 - Journal of Symbolic Logic 79 (1):103-134.