Notre Dame Journal of Formal Logic
- Notre Dame J. Formal Logic
- Volume 60, Number 2 (2019), 195-214.
In this article, we develop and clarify some of the basic combinatorial properties of the new notion of -dependence (for ) 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, -dependence corresponds to the inability to encode a random -partite -hypergraph with a definable edge relation. We characterize -dependence by counting -types over finite sets (generalizing the Sauer–Shelah lemma, answering a question of Shelah), and in terms of the collapse of random ordered -hypergraph indiscernibles down to order-indiscernibles (which implies that the failure of -dependence is always witnessed by a formula in a single free variable).
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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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. https://projecteuclid.org/euclid.ndjfl/1557129619