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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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  • 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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  • On Positive Local Combinatorial Dividing-Lines in Model Theory.Vincent Guingona & Cameron Donnay Hill - 2019 - Archive for Mathematical Logic 58 (3-4):289-323.
    We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-of-indiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.
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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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  • 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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  • Keisler’s Order is Not Linear, Assuming a Supercompact.Douglas Ulrich - 2018 - Journal of Symbolic Logic 83 (2):634-641.
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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.