11 found
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  1.  44
    Theories without the tree property of the second kind.Artem Chernikov - 2014 - Annals of Pure and Applied Logic 165 (2):695-723.
    We initiate a systematic study of the class of theories without the tree property of the second kind — NTP2. Most importantly, we show: the burden is “sub-multiplicative” in arbitrary theories ; NTP2 is equivalent to the generalized Kimʼs lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters — so the dp-rank of a 1-type in any theory is (...)
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  2.  33
    On model-theoretic tree properties.Artem Chernikov & Nicholas Ramsey - 2016 - Journal of Mathematical Logic 16 (2):1650009.
    We study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are (...)
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  3.  66
    Forking and dividing in NTP₂ theories.Artem Chernikov & Itay Kaplan - 2012 - Journal of Symbolic Logic 77 (1):1-20.
    We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2.
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  4.  47
    On $n$ -Dependence.Artem Chernikov, Daniel Palacin & Kota Takeuchi - 2019 - Notre Dame Journal of Formal Logic 60 (2):195-214.
    In this article, we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence recently introduced by Shelah. In the same way as dependence of a theory means its inability to encode a bipartite random graph with a definable edge relation, n-dependence corresponds to the inability to encode a random -partite -hypergraph with a definable edge relation. We characterize n-dependence by counting φ-types over finite sets, and in terms of the collapse of random ordered -hypergraph (...)
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  5.  30
    An independence theorem for ntp2 theories.Itaï Ben Yaacov & Artem Chernikov - 2014 - Journal of Symbolic Logic 79 (1):135-153.
  6.  18
    Henselian valued fields and inp-minimality.Artem Chernikov & Pierre Simon - 2019 - Journal of Symbolic Logic 84 (4):1510-1526.
    We prove that every ultraproduct of p-adics is inp-minimal. More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.
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  7.  20
    Semi-Equational Theories.Artem Chernikov & Alex Mennen - forthcoming - Journal of Symbolic Logic:1-32.
    We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.
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  8.  20
    Transitivity, Lowness, and Ranks in Nsop Theories.Artem Chernikov, K. I. M. Byunghan & Nicholas Ramsey - 2023 - Journal of Symbolic Logic 88 (3):919-946.
    We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $_{1}$ theories.
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  9.  21
    Invariant measures in simple and in small theories.Artem Chernikov, Ehud Hrushovski, Alex Kruckman, Krzysztof Krupiński, Slavko Moconja, Anand Pillay & Nicholas Ramsey - 2023 - Journal of Mathematical Logic 23 (2).
    We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups (...)
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  10.  20
    Model-theoretic Elekes–Szabó in the strongly minimal case.Artem Chernikov & Sergei Starchenko - 2020 - Journal of Mathematical Logic 21 (2):2150004.
    We prove a generalization of the Elekes–Szabó theorem [G. Elekes and E. Szabó, How to find groups?, Combinatorica 32 537–571 ] for relations defina...
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  11.  28
    Model theory, Keisler measures, and groups - Ehud Hrushovski, Ya’acov Peterzil and Anand Pillay, Groups, measures, and the NIP. Journal of the American Mathematical Society, vol. 21 , no. 2, pp. 563–596. - Ehud Hrushovski and Anand Pillay, On NIP and invariant measures. Journal of the European Mathematical Society, vol.13 , no. 4, pp. 1005–1061. - Ehud Hrushovski, Anand Pillay, and Pierre Simon, Generically stable and smooth measures in NIP theories. Transactions of the American Mathematical Society, vol. 365 , no. 5, pp. 2341–2366. [REVIEW]Artem Chernikov - 2018 - Bulletin of Symbolic Logic 24 (3):336-339.
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