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Preprint Number 791
791. Artem Chernikov, Daniel Palacin and Kota Takeuchi
Submission date: 1 November 2014.
In this note we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1 \leq n < 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. Most importantly, we characterize n-dependence by counting φ-types over finite sets (generalizing Sauer-Shelah lemma and 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).
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