Persistence and NIP in the characteristic sequence

TL;DR

This paper introduces the characteristic sequence for a first-order formula and explores how its combinatorial properties reflect the formula's classification theoretic features, revealing connections between hypergraph configurations and model-theoretic properties.

Contribution

It establishes a link between the combinatorial structure of the characteristic sequence and the classification theoretic properties of formulas, including stability and independence.

Findings

Tree properties are detected by specific configurations in the sequence.

Instability manifests as persistent complex configurations under localization.

The characteristic sequence encodes key model-theoretic properties.

Abstract

For a first-order formula $\phi(x;y)$ we introduce and study the characteristic sequence $<P_n : n < \omega>$ of hypergraphs defined by $P_n(y_1,...,y_n) := (\exists x) \bigwedge_{i \leq n} \phi(x;y_i)$. We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of $\phi$ and vice versa. Specifically, we show that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization.