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On n-dependence

Artem Chernikov, Daniel Palacin, Kota Takeuchi

Abstract

In this note we develop and clarify some of the basic combinatorial properties of the new notion of $n$-dependence (for $1\leq n < ω$) 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 $\varphi$-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).

On n-dependence

Abstract

In this note 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. Most importantly, we characterize -dependence by counting -types over finite sets (generalizing Sauer-Shelah lemma and 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).

Paper Structure

This paper contains 13 sections, 21 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. $n$-dependence
  3. Counting $\varphi$-types and a generalization of Sauer-Shelah lemma
  4. Sauer-Shelah Lemma and Generalized VC-dimension
  5. $VC_n$-dimension and $n$-dependence
  6. Generalized indiscernibles
  7. Ramsey property and hypergraphs
  8. Generalized indiscernibles
  9. ${\rm IP}_{n}$ and random hypergraph indiscernibles
  10. Basic properties of ${\rm IP}_n$ and indiscernible witnesses
  11. Characterizations of ${\rm NIP}_n$ by collapsing indiscernibles
  12. Reduction to 1 variable
  13. Ramsey property for hypergraphs

Key Result

Proposition \oldthetheorem

Let ${\cal C}$ be a class of subsets of $X$.

Theorems & Definitions (57)

  • Definition \oldthetheorem
  • Example \oldthetheorem
  • Definition \oldthetheorem: Vapnik and Chervonenkis
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof : Proof of Proposition \ref{['generalized Sauer']}
  • Claim
  • proof : Proof of the claim
  • ...and 47 more