Notre Dame Journal of Formal Logic

On n-Dependence

Artem Chernikov, Daniel Palacin, and Kota Takeuchi

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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).

Article information

Notre Dame J. Formal Logic, Volume 60, Number 2 (2019), 195-214.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C45: Classification theory, stability and related concepts [See also 03C48]
Secondary: 05C55: Generalized Ramsey theory [See also 05D10]

n-dependence Sauer–Shelah lemma generalized indiscernibles structural Ramsey theory


Chernikov, Artem; Palacin, Daniel; Takeuchi, Kota. On $n$ -Dependence. Notre Dame J. Formal Logic 60 (2019), no. 2, 195--214. doi:10.1215/00294527-2019-0002.

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