Characterizations of monadic NIP

TL;DR

This paper characterizes when a complete first-order theory is monadically NIP, focusing on types, model decompositions, indiscernibles, and forbidden configurations, with applications to hereditary classes of finite structures.

Contribution

It provides multiple new characterizations of monadically NIP theories, including a key condition on finite satisfiability of types and model decompositions.

Findings

Characterizations based on finite satisfiability of types.

Decomposition of models and behavior of indiscernibles.

Non-structure results for hereditary classes of finite substructures.

Abstract

We give several characterizations of when a complete first-order theory $T$ is monadically NIP, i.e. when expansions of $T$ 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.