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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Externally definable sets and dependent pairs II
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by Artem Chernikov and Pierre Simon PDF
Trans. Amer. Math. Soc. 367 (2015), 5217-5235 Request permission

Abstract:

We continue investigating the structure of externally definable sets in $\mathrm {NIP}$ theories and preservation of $\mathrm {NIP}$ after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in a distal theory; naming an arbitrary small indiscernible sequence preserves $\mathrm {NIP}$, while naming a large one doesn’t; there are models of $\mathrm {NIP}$ theories over which all 1-types are definable, but not all $n$-types.
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Additional Information
  • Artem Chernikov
  • Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, 91904 Jerusalem, Israel
  • Address at time of publication: Équipe de Logique Mathématique, IMJ - PRG, Université Paris Diderot Paris 7, UFR de Mathématiques, case 7012, 75205 Paris Cedex 13, France
  • Email: artem.chernikov@imj-prg.fr
  • Pierre Simon
  • Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, 91904 Jerusalem, Israel
  • Address at time of publication: Université Claude Bernard - Lyon 1, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
  • MR Author ID: 942320
  • Email: simon@math.univ-lyon1.fr
  • Received by editor(s): April 10, 2012
  • Received by editor(s) in revised form: May 16, 2013
  • Published electronically: February 20, 2015
  • Additional Notes: The first author was supported by the Marie Curie Initial Training Network in Mathematical Logic - MALOA - From Mathematical Logic to Applications, PITN-GA-2009-238381
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 367 (2015), 5217-5235
  • MSC (2010): Primary 03C45, 03C50; Secondary 05D99, 68R05
  • DOI: https://doi.org/10.1090/S0002-9947-2015-06210-2
  • MathSciNet review: 3335415