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  1. 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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  • Generic Variations and NTP$$_1$$1.Jan Dobrowolski - 2018 - Archive for Mathematical Logic 57 (7-8):861-871.
    We prove a preservation theorem for NTP\ in the context of the generic variations construction. We also prove that NTP\ is preserved under adding to a geometric theory a generic predicate.
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  • 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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  • Characterization of NIP Theories by Ordered Graph-Indiscernibles.Lynn Scow - 2012 - Annals of Pure and Applied Logic 163 (11):1624-1641.
    We generalize the Unstable Formula Theorem characterization of stable theories from Shelah [11], that a theory T is stable just in case any infinite indiscernible sequence in a model of T is an indiscernible set. We use a generalized form of indiscernibles from [11], in our notation, a sequence of parameters from an L-structure M, , indexed by an L′-structure I is L′-generalized indiscernible inM if qftpL′=qftpL′ implies tpL=tpL for all same-length, finite ¯,j from I. Let Tg be the theory (...)
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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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  • A Note on Nsop1 in One Variable.Nicholas Ramsey - 2019 - Journal of Symbolic Logic 84 (1):388-392.
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  • Tree Indiscernibilities, Revisited.Byunghan Kim, Hyeung-Joon Kim & Lynn Scow - 2014 - Archive for Mathematical Logic 53 (1-2):211-232.
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  • 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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  • 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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  • 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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  • 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 Ultrafilters Built by Independent Families of Functions.M. Malliaris & S. Shelah - 2014 - Journal of Symbolic Logic 79 (1):103-134.