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Externally definable sets and dependent pairs II


Authors: Artem Chernikov and Pierre Simon
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
Published electronically: February 20, 2015
MathSciNet review: 3335415
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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
Email: simon@math.univ-lyon1.fr

DOI: https://doi.org/10.1090/S0002-9947-2015-06210-2
Keywords: NIP, UDTFS, externally definable sets, VC-dimension, elementary pairs
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
Article copyright: © Copyright 2015 American Mathematical Society