Artem Chernikov

Theories without the tree property of the second kind

Abstract 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 Kimʼs 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.

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.

Externally definable sets and dependent pairs II

© 2015 American Mathematical Society. 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.

On model-theoretic tree properties

We study model theoretic tree properties (

An independence theorem for ntp2 theories

\text{TP}, \text{TP}_1, \text{TP}_2

ON NON-FORKING SPECTRA

) and their associated cardinal invariants (

On the number of dedekind cuts and two-cardinal models of dependent theories

\kappa_{\text{cdt}}, \kappa_{\text{sct}}, \kappa_{\text{inp}}

Regularity lemma for distal structures

, respectively). In particular, we obtain a quantitative refinement of Shelahs theorem (

Definably amenable NIP groups

\text{TP} \Rightarrow \text{TP}_1 \lor \text{TP}_2

Groups and fields with NTP2

) for countable theories, show that