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).
"On -Dependence." Notre Dame J. Formal Logic 60 (2) 195 - 214, May 2019. https://doi.org/10.1215/00294527-2019-0002