Externally definable sets and dependent pairs II

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.

Imaginaries and invariant types in existentially closed valued differential fields

“…By Theorem (A.5), it suffices to show that R is uniformly stably embedded (as an ordered group) in any elementary extension. But that follows from the fact that (R, <) is complete and (R, +, <) is ominimal, see [CS15,Corollary 64]. ∎ Remark A.8: An easy consequence of this result is that the constant field C K is stably embedded in models of VDF EC .…”

Dense codense predicates and the NTP2

“…There are several papers on the preservation of stability and the NIP under adding predicates. For example, the work of Casanovas and Ziegler on stability and a similar result by Chernikov and Simon on NIP theories . A key fact in both papers is the following result: inside a highly saturated model of T , the family of stable (or NIP) formulas is closed under Boolean combination and it suffices to check (under some technical assumptions) that the induced structure on a predicate is stable (has the NIP).…”

External definability and groups in NIP theories

“…Although real Lie groups are outside the domain of stability, they are, more or less, groups definable in o-minimal theories, and have been studied by model theorists from this point of view for some time. A common Examples of structures satisfying the assumption of the theorem are any model of a stable theory, (R, +, ·), (Q p , +, ·), (Z, +, <) (see [4,Section 5] for a discussion of this phenomenon).…”

“…(i) The projection of an externally definable subset of M is externally definable. (ii) In particular, Th(M ext ) eliminates quantifiers, and is NIP.Further study of externally definable sets in NIP theories, as well as a refined and uniform version of Shelah's theorem, can be found in [3,4].So one aim of this paper is to show that many properties of (for example, definable amenability) and objects attached to (for example, G 00 ) a group G definable over a model M of an NIP theory T are preserved when passing to Th(M ext ), answering some questions raised in [8]. A second aim of this paper, bearing in mind the above, is to prove some more cases of the 'Ellis group' conjecture (originating with Newelski) which says that in the NIP environment, for suitable groups G definable over a model M , G/G 00 should coincide with the 'Ellis group' computed in Th(M ext ), where all types over M ext are definable.…”

“…Further study of externally definable sets in NIP theories, as well as a refined and uniform version of Shelah's theorem, can be found in [3,4].…”