On n-Dependence

Abstract

In this article, we develop and clarify some of the basic combinatorial properties of the new notion of n-dependence (for 1≤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. 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).