Distal and non-distal NIP theories

TL;DR

This paper explores the concept of distality in NIP theories, defining it as a form of pure instability, and investigates the structure and properties of theories when distality fails, including stable parts of types and quantifier elimination.

Contribution

It introduces the notion of distality in NIP theories, analyzes the stable components when distality fails, and applies these ideas to quantifier elimination in model expansions.

Findings

Distality characterizes stable phenomena in NIP theories.

Failure of distality leads to the extraction of stable parts of types.

Expansion by traces of externally definable sets eliminates quantifiers.

Abstract

We study one way in which stable phenomena can exist in an NIP theory. We start by defining a notion of 'pure instability' that we call 'distality' in which no such phenomenon occurs. O-minimal theories and the p-adics for example are distal. Next, we try to understand what happens when distality fails. Given a type p over a sufficiently saturated model, we extract, in some sense, the stable part of p and define a notion of stable-independence which is implied by non-forking and has bounded weight. As an application, we show that the expansion of a model by traces of externally definable sets from some adequate indiscernible sequence eliminates quantifiers.