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arXiv · 1411.0120

On n-dependence

Artem Chernikov·Daniel Palacin·Kota Takeuchi
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Abstract

In this note we develop and clarify some of the basic combinatorial properties of the new notion of $n$-dependence (for $1\leq n < \omega$) 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 $\varphi$-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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Artem Chernikov, Daniel Palacin, Kota Takeuchi. 2014-11-01. On n-dependence. https://doi.org/10.1215/00294527-2019-0002

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