Abstract

We give several characterizations of when a complete first-order theory T T is monadically NIP, i.e. when expansions of T T by arbitrary unary predicates do not have the independence property. The central characterization is a condition on finite satisfiability of types. Other characterizations include decompositions of models, the behavior of indiscernibles, and a forbidden configuration. As an application, we prove non-structure results for hereditary classes of finite substructures of non-monadically NIP models that eliminate quantifiers.

Highlights

  • It is well known that many first-order theories whose models are tame can become unwieldy after naming a unary predicate

  • As part of a larger project in [2], Baldwin and Shelah undertook a study of this phenomenon. They found that a primary dividing line is whether T admits coding i.e., there are three subsets A, B, C of a model of T and a formula φ(x, y, z) that defines a pairing function A × B → C

  • The primary focus in [2] was monadically stable theories, i.e. theories that remain stable after arbitrary expansions by unary predicates

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Summary

Introduction

It is well known that many first-order theories whose models are tame can become unwieldy after naming a unary predicate. Applying Lemma 2.20 to I (in the theory TD naming constants for each d ∈ D) choose a model M ⊇ D and a full set C ⊇ M such that I is both indiscernible over C and an M -f.s. sequence over C. As tp( ̄bs/Das) is finitely satisfied in D and D is full, by Lemma 2.13 a

The main theorem
Finite structures
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