Remarks on the NIP in a model

Abstract

We define the notion φ(x,y) has the NIP (not the independence property) in A, where A is a subset of a model, and give some equivalences by translating results from function theory. We also discuss the number of coheirs when A is not necessarily countable, and revisit the notion “φ(x,y) has the NOP (not the order property) in a model M”.

Stability, the NIP, and the NSOP: model theoretic properties of formulas via topological properties of function spaces

“…Remark Note that one can say more: a formula φ is NIP on M if and only if every coheir is Baire 1 definable. This is discussed in detail in [16]. …”

Local Keisler Measures and Nip Formulas

“…From this vantage point, we can apply an important result of Bourgain, Fremlin, and Talagrand [2], namely Theorem 2.5. The connection between NIP formulas, local types, and Theorem 2.5 was first noticed by Simon in [13] and further work has been done by Khanaki and Pillay in [9] and [10]. We can further extend this connection to local measures via Ben Yaacov's work on continuous VC classes [1].…”

“…The connection between NIP formulas and Theorem 2.5 was first noticed by Simon and Chernikov in [3] as well as independently by Ibarlucía in [10]. Furthermore work extending this connection has been done by Simon in [16] as well as the NIP in a model case by Khanaki and Pillay in [12] and [13]. We extend this connection to local measures via Ben Yaacov's work on continuous VC classes [1].…”

“…In fact, a much more general statement is proven in [2] than the one we provide. The connection between this theorem and NIP formulas is well known and has been expanded upon in [13], [9], and [10]. This theorem is slightly more technical than the last two, but essentially it allows one to find pointwise convergent sequences under particular tameness conditions.…”

Galois groups as quotients of Polish groups

“…In this section, we discuss the relationship between model-theoretic NIP and dynamical tameness. A relationship between the Bourgain-Fremlin-Talagrand dichotomy and NIP seems to have been first noticed independently in [CS18], [Iba16], and [Kha14]; see also [Sim15] and [KP17a] for related research. Many statements in this section appear to be folklore, but we have not found them stated and proved in this form, so we present them along with their proofs, as they are interesting in their own right.…”