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The characteristic sequence of a first-order formula

  • M. E. Malliaris (a1)

For a first-order formula φ(x; y) we introduce and study the characteristic sequence (Pn: n < ω) of hypergraphs defined by . 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 some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization.

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[2]Džamonja and Shelah, On ⊲*-maximality, Annals of Pure and Applied Logic, vol. 125 (2004), pp. 119158.

[3]Hrushovski, Peterzil, and Pillay, Groups, measures, and the NIP, Journal of the American Mathematical Society, vol. 21 (2008), no. 2, pp. 563596.

[7]Malliaris, Realization of φ-types and Keisler's order, Annals of Pure and Applied Logic, vol. 157 (2009), pp. 220224.

[9]Shelah, Toward classifying unstable theories, Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229255.

[12]Shelah and Usvyatsov, More on SOP1and SOP2, Annals of Pure and Applied Logic, vol. 155 (2008), no. 1, pp. 1631.

[14]Wagner, Simple theories, Mathematics and Its Applications 503, Kluwer Academic Publishers, 2000.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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