NIP for some pair-like theories

Abstract

Abstract. Generalising work from [2] and [6], we give sufficient conditions for a theory TP to inherit N IP from T , where TP is an expansion of the theory T by a unary predicate P . We apply our result to theories, studied in [1], of the real field with a subgroup of the unit circle.

Externally definable sets and dependent pairs

“…dp + -minimal) when for a singleton c there is such a u of size 1. This was already proved by Berenstein, Dolich and Onshuus in [BDO08] and generalised by Boxall in [Box09]. Our result generalises [BDO08, Theorem 2.7], since the hypothesis there (acl is a pregeometry and A is "innocuous") imply boundedness of T P .…”

“…In the last few years there has been a large number of papers proving dependence for some pair-like structures, e.g. [BDO08], [GH10], [Box09], etc. We apologise for adding yet another result to the list.…”

The independence property in generalized dense pairs of structures

“…The aim of Section 3 is to show that if the original theory is (strongly) dependent, then the corresponding theory of lovely pairs is again (strongly) dependent. Similar results have been obtained independently by Boxall [5] and by Giinaydin and Hieronymi [16]. For this section we assume the reader is familiar with basic ideas of rosy theories and dependent theories, we refer the reader to [1, 20] and [21].…”

Distal and Non-Distal Pairs

“…All the above examples are NIP. For dense pairs this is due independently to Berenstein, Dolich, Onshuus [2], Boxall [3], and Günaydın and Hieronymi [12]; for dense groups this was shown in [3] and [12]; for tame pairs and for the discrete subgroups NIP was first proven in [12]. For a later, but more general result implying NIP for all these theories, see Chernikov and Simon [4].…”