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  1. 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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  • Club guessing sequences and filters.Tetsuya Ishiu - 2005 - Journal of Symbolic Logic 70 (4):1037-1071.
    We investigate club guessing sequences and filters. We prove that assuming V=L, there exists a strong club guessing sequence on μ if and only if μ is not ineffable for every uncountable regular cardinal μ. We also prove that for every uncountable regular cardinal μ, relative to the existence of a Woodin cardinal above μ, it is consistent that every tail club guessing ideal on μ is precipitous.
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  • 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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  • A Tail Club Guessing Ideal Can Be Saturated without Being a Restriction of the Nonstationary Ideal.Tetsuya Ishiu - 2005 - Notre Dame Journal of Formal Logic 46 (3):327-333.
    We outline the proof of the consistency that there exists a saturated tail club guessing ideal on ω₁ which is not a restriction of the nonstationary ideal. A new class of forcing notions and the forcing axiom for the class are introduced for this purpose.
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  • Divide and Conquer: Dividing Lines and Universality.Saharon Shelah - 2021 - Theoria 87 (2):259-348.
    We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results.
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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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  • 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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  • Compactly expandable models and stability.Enrique Casanovas - 1995 - Journal of Symbolic Logic 60 (2):673-683.
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  • The Dividing Line Methodology: Model Theory Motivating Set Theory.John T. Baldwin - 2021 - Theoria 87 (2):361-393.
    We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to clarify the “main gap” by reducing the (...)
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