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Universal graphs at the successor of a singular cardinal

Mirna D.?Amonja & Saharon Shelah

Journal of Symbolic Logic 68 (2): 366- 388 (2003)

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On ◁∗-maximality.Mirna Džamonja & Saharon Shelah - 2004 - Annals of Pure and Applied Logic 125 (1-3):119-158.details 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 (...) Download Export citation Bookmark 31 citations
The descriptive set-theoretical complexity of the embeddability relation on models of large size.Luca Motto Ros - 2013 - Annals of Pure and Applied Logic 164 (12):1454-1492.details We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for (...) Download Export citation Bookmark 5 citations
Divide and Conquer: Dividing Lines and Universality.Saharon Shelah - 2021 - Theoria 87 (2):259-348.details 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. Download Export citation Bookmark 1 citation
Small universal families for graphs omitting cliques without GCH.Katherine Thompson - 2010 - Archive for Mathematical Logic 49 (7-8):799-811.details When no single universal model for a set of structures exists at a given cardinal, then one may ask in which models of set theory does there exist a small family which embeds the rest. We show that for λ+-graphs (λ regular) omitting cliques of some finite or uncountable cardinality, it is consistent that there are small universal families and 2λ > λ+. In particular, we get such a result for triangle-free graphs. Download Export citation Bookmark
Cardinal characteristics at κ in a small u ( κ ) model.A. D. Brooke-Taylor, V. Fischer, S. D. Friedman & D. C. Montoya - 2017 - Annals of Pure and Applied Logic 168 (1):37-49.details Download Export citation Bookmark 4 citations
The descriptive set-theoretical complexity of the embeddability relation on models of large size.Luca Ros - 2013 - Annals of Pure and Applied Logic 164 (12):1454-1492.details We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for (...) Download Export citation Bookmark 5 citations
Ramsey Theory for Countable Binary Homogeneous Structures.Jean A. Larson - 2005 - Notre Dame Journal of Formal Logic 46 (3):335-352.details Countable homogeneous relational structures have been studied by many people. One area of focus is the Ramsey theory of such structures. After a review of background material, a partition theorem of Laflamme, Sauer, and Vuksanovic for countable homogeneous binary relational structures is discussed with a focus on the size of the set of unavoidable colors. Download Export citation Bookmark 1 citation
Models of real-valued measurability.Sakae Fuchino, Noam Greenberg & Saharon Shelah - 2006 - Annals of Pure and Applied Logic 142 (1):380-397.details Solovay’s random-real forcing [R.M. Solovay, Real-valued measurable cardinals, in: Axiomatic Set Theory , Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428] is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measure-theoretic properties of Solovay’s model that do not follow from the existence of real-valued measurability. Download Export citation Bookmark 1 citation
Universal graphs at ℵ ω 1 + 1.Jacob Davis - 2017 - Annals of Pure and Applied Logic 168 (10):1878-1901.details Download Export citation Bookmark

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