Characterizations of monadic NIP

TL;DR

The paper develops a comprehensive set of characterizations for monadic NIP theories by isolating a central finite-satisfiability dichotomy (the f.s. dichotomy) and linking it to model decompositions, indiscernible behavior, and coding configurations. It proves the equivalence of six conditions, notably that monadic NIP coincides with the f.s. dichotomy, extendable $M$-f.s. decompositions, dp-minimality with endless indiscernible triviality, and the absence of coding on tuples or in monadic expansions. A key methodological contribution is the analysis of $M$-f.s. sequences and their relation to indiscernibles, enabling a linearized decomposition approach akin to tree decompositions in stable theories. The paper also connects these model-theoretic properties to the combinatorics of hereditary classes via Age$(T)$, showing that monadic NIP aligns with NIP/stability of Age$(T)$ under quantifier elimination and yields non-structure results when not monadically NIP, including growth-rate and well-quasi-order considerations.

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.

Theorems & Definitions (109)

• Theorem 1.1
• Conjecture 1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.4: $M$-f.s. sequence
• Definition 2.5
• Lemma 2.6
• proof