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  1. Dense Codense Predicates and the NTP2.Alexander Berenstein & Hyeung-Joon Kim - 2016 - Mathematical Logic Quarterly 62 (1-2):16-24.
  • 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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  • 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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  • 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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