On n-dependence

TL;DR

This paper explores the combinatorial properties of n-dependence in model theory, generalizing dependence to higher dimensions and characterizing it through types and indiscernibles.

Contribution

It provides a characterization of n-dependence via counting types and hypergraph indiscernibles, extending Shelah's work and answering a key question.

Findings

n-dependence corresponds to the inability to encode certain hypergraphs

Characterization of n-dependence by counting types over finite sets

Collapse of hypergraph indiscernibles to order-indiscernibles

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).