Externally definable sets and dependent pairs II

TL;DR

This paper advances the understanding of externally definable sets in NIP theories, exploring how expansions by new predicates affect properties like definability, non-forking, and NIP preservation, with several criteria and examples provided.

Contribution

It introduces new results on uniform definability of types, criteria for boundedness in distal theories, and effects of naming indiscernible sequences on NIP.

Findings

Types over finite sets are uniformly definable.

Finitely many invariant types cover non-forking instances over models.

Naming small indiscernible sequences preserves NIP, large ones do not.

Abstract

We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in a distal theory; naming an arbitrary small indiscernible sequence preserves NIP, while naming a large one doesn't; there are models of NIP theories over which all 1-types are definable, but not all n-types.