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  1. Divide and Conquer: Dividing Lines and Universality.Saharon Shelah - 2021 - Theoria 87 (2):259-348.
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  • The Dividing Line Methodology: Model Theory Motivating Set Theory.John T. Baldwin - 2021 - Theoria 87 (2):361-393.
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  • Atomic Saturation of Reduced Powers.Saharon Shelah - forthcoming - Mathematical Logic Quarterly.
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  • Keisler’s order via Boolean ultrapowers.Francesco Parente - 2021 - Archive for Mathematical Logic 60 (3):425-439.
    In this paper, we provide a new characterization of Keisler’s order in terms of saturation of Boolean ultrapowers. To do so, we apply and expand the framework of ‘separation of variables’ recently developed by Malliaris and Shelah. We also show that good ultrafilters on Boolean algebras are precisely the ones which capture the maximum class in Keisler’s order, answering a question posed by Benda in 1974.
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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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  • 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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  • On Reduced Products and Filters.Mroslav Benda - 1972 - Annals of Mathematical Logic 4 (1):1.
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  • Saturation of Ultrapowers and Keisler's Order.Saharon Shelah - 1972 - Annals of Mathematical Logic 4 (1):75.
  • Stability, the F.C.P., and Superstability; Model Theoretic Properties of Formulas in First Order Theory.Saharon Shelah - 1971 - Annals of Mathematical Logic 3 (3):271.
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  • The Metamathematics of Random Graphs.John T. Baldwin - 2006 - Annals of Pure and Applied Logic 143 (1-3):20-28.
    We explain and summarize the use of logic to provide a uniform perspective for studying limit laws on finite probability spaces. This work connects developments in stability theory, finite model theory, abstract model theory, and probability. We conclude by linking this context with work on the Urysohn space.
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  • Finite Diagrams Stable in Power.Saharon Shelah - 1970 - Annals of Mathematical Logic 2 (1):69.
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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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  • 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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  • Saturation and Simple Extensions of Models of Peano Arithmetic.Matt Kaufmann & James H. Schmerl - 1984 - Annals of Pure and Applied Logic 27 (2):109-136.
  • Weakly Minimal Formulas: A Global Approach.Ludomir Newelski - 1990 - Annals of Pure and Applied Logic 46 (1):65-94.
  • A New Look at Interpretability and Saturation.M. Malliaris & S. Shelah - 2019 - Annals of Pure and Applied Logic 170 (5):642-671.
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