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May 2019 On n-Dependence
Artem Chernikov, Daniel Palacin, Kota Takeuchi
Notre Dame J. Formal Logic 60(2): 195-214 (May 2019). DOI: 10.1215/00294527-2019-0002

Abstract

In this article, we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1n<ω) 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 (n+1)-partite (n+1)-hypergraph with a definable edge relation. We characterize n-dependence by counting φ-types over finite sets (generalizing the Sauer–Shelah lemma, answering a question of Shelah), and in terms of the collapse of random ordered (n+1)-hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of n-dependence is always witnessed by a formula in a single free variable).

Citation

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Artem Chernikov. Daniel Palacin. Kota Takeuchi. "On n-Dependence." Notre Dame J. Formal Logic 60 (2) 195 - 214, May 2019. https://doi.org/10.1215/00294527-2019-0002

Information

Received: 25 November 2015; Accepted: 13 December 2016; Published: May 2019
First available in Project Euclid: 6 May 2019

zbMATH: 07096536
MathSciNet: MR3952231
Digital Object Identifier: 10.1215/00294527-2019-0002

Subjects:
Primary: 03C45
Secondary: 05C55

Rights: Copyright © 2019 University of Notre Dame

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Vol.60 • No. 2 • May 2019
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