Linear orders in NIP structures

TL;DR

This paper proves that unstable NIP theories have definable linear quasi-orders, and omega-categorical ones interpret infinite linear orders, addressing a longstanding open question in model theory.

Contribution

It establishes the existence of definable linear quasi-orders in unstable NIP theories and shows omega-categorical theories interpret infinite linear orders, advancing understanding of NIP structures.

Findings

Unstable NIP theories admit V-definable linear quasi-orders.

Omega-categorical NIP theories interpret infinite linear orders.

Partially answers a longstanding open question in model theory.

Abstract

We show that every unstable NIP theory admits a V-definable linear quasi-order, over a finite set of parameters. In particular, if the theory is omega-categorical, then it interprets an infinite linear order. This partially answers a longstanding open question.