Externally definable sets and dependent pairs II

Abstract

We continue investigating the structure of externally definable sets in N I P \mathrm {NIP} theories and preservation of N I P \mathrm {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 N I P \mathrm {NIP} , while naming a large one doesn’t; there are models of N I P \mathrm {NIP} theories over which all 1-types are definable, but not all n n -types.