Edge distribution and density in the characteristic sequence

TL;DR

This paper explores how graph-theoretic tools like Szemerédi's regularity lemma can analyze the complexity of formulas in model theory through the characteristic sequence of hypergraphs, linking combinatorics and logic.

Contribution

It demonstrates the application of graph regularity techniques to classify model-theoretic properties and analyzes independence and order properties in unstable theories.

Findings

Relates classification properties of formulas to graph densities in regular decompositions.

Uses Szemerédi regularity to measure the depth of independence among sets.

Connects failures of independence depth to Shelah's SOP_3 property.

Abstract

The characteristic sequence of hypergraphs $<P_n : n<\omega>$ associated to a formula $\phi(x;y)$, introduced in [arXiv:0908.4111], is defined by $P_n(y_1,... y_n) = (\exists x) \bigwedge_{i\leq n} \phi(x;y_i)$. This paper continues the study of characteristic sequences, showing that graph-theoretic techniques, notably Szemer\'edi's celebrated regularity lemma, can be naturally applied to the study of model-theoretic complexity via the characteristic sequence. Specifically, we relate classification-theoretic properties of $\phi$ and of the $P_n$ (considered as formulas) to density between components in Szemer\'edi-regular decompositions of graphs in the characteristic sequence. In addition, we use Szemer\'edi regularity to calibrate model-theoretic notions of independence by describing the depth of independence of a constellation of sets and showing that certain failures of depth imply Shelah's strong order property $SOP_3$; this sheds light on the interplay of independence and order in unstable theories.