Stable embeddedness and NIP

We give sufficient conditions for a predicate P in a complete theory T to be stably embedded: P with its induced 0-definable structure has "finite rank", P has NIP in T and P is 1-stably embedded. This generalizes recent work by Hasson and Onshuus in the case where P is o-minimal in T.Comment: 10 page

Externally definable sets and dependent pairs

We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah's expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.Comment: 17 pages, some typos and mistakes corrected, overall presentation improved, more details for the examples are give

Henselian valued fields and inp-minimality

We prove that every ultraproduct of $p$-adics is inp-minimal (i.e., of burden $1$). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic $0$ in the RV language.Comment: v.2: 15 pages, minor corrections and presentation improvements; accepted to the Journal of Symbolic Logi

Definable and invariant types in enrichments of NIP theories

Let T be an NIP L-theory and T' be an enrichment. We give a sufficient condition on T' for the underlying L-type of any definable (respectively invariant) type over a model of T' to be definable (respectively invariant) as an L-type. Besides, we generalise work of Simon and Starchenko on the density of definable types among non forking types to this relative setting. These results are then applied to Scanlon's model completion of valued differential fields.Comment: 9 pages. An error was pointed out in section 2 of the previous version so that section was removed. So was Proposition 3.8 that depended on i

Theories without the tree property of the second kind

We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is "sub-multiplicative" in arbitrary theories (in particular, if a theory has TP2 then there is a formula with a single variable witnessing this); NTP2 is equivalent to the generalized Kim's lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters - so the dp-rank of a 1-type in any theory is always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic (0,0) is NTP2 (strong, of finite burden) if and only if the residue field is NTP2 (the residue field and the value group are strong, of finite burden respectively), so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2.Comment: 35 pages; v.3: a discussion and a Conjecture 2.7 on the sub-additivity of burden had been added; Section 3.1 on the SOPn hierarchy restricted to NTP2 theories had been added; Problem 7.13 had been updated; numbering of theorems had been changed and some minor typos were fixed; Annals of Pure and Applied Logic, accepte

Imaginaries in separably closed valued fields

We show that separably closed valued fields of finite imperfection degree (either with lambda-functions or commuting Hasse derivations) eliminate imaginaries in the geometric language. We then use this classification of interpretable sets to study stably dominated types in those structures. We show that separably closed valued fields of finite imperfection degree are metastable and that the space of stably dominated types is strict pro-definable