Itay Kaplan

Additivity of the dp-rank

The main result is the prove of the linearity of the dp-rank. We also prove that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types and discuss the possible relations between dp-rank and VC-density.

Chain conditions in dependent groups

Abstract In this note we prove and disprove some chain conditions in type definable and definable groups in dependent, strongly dependent and strongly 2 dependent theories.

ON NON-FORKING SPECTRA

Non-forking is one of the most important notions in modern model theory capturing the idea of a generic extension of a type (which is a far-reaching generalization of the concept of a generic point of a variety). To a countable first-order theory we associate its non-forking spectrum - a function of two cardinals kappa and lambda giving the supremum of the possible number of types over a model of size lambda that do not fork over a sub-model of size kappa. This is a natural generalization of the stability function of a theory. We make progress towards classifying the non-forking spectra. On the one hand, we show that the possible values a non-forking spectrum may take are quite limited. On the other hand, we develop a general technique for constructing theories with a prescribed non-forking spectrum, thus giving a number of examples. In particular, we answer negatively a question of Adler whether NIP is equivalent to bounded non-forking. In addition, we answer a question of Keisler regarding the number of cuts a linear order may have. Namely, we show that it is possible that ded(kappa) < ded(kappa)^omega.

The Borel cardinality of Lascar strong types

We show that if the restriction of the Lascar equivalence relation to a KP-strong type is non-trivial, then it is non-smooth (when viewed as a Borel equivalence relation on an appropriate space of types).

DECIDABILITY AND CLASSIFICATION OF THE THEORY OF INTEGERS WITH PRIMES

We show that under Dicksons conjecture about the distribution of primes in the natural numbers, the theory Th(Z,+,1,0,Pr) where Pr is a predicate for the prime numbers and their negations is decidable, unstable and supersimple. This is in contrast with Th(Z,+,0,Pr,<) which is known to be undecidable by the works of Jockusch, Bateman and Woods.

Some Remarks on dp-Minimal Groups

We prove that ω-categorical dp-minimal groups are nilpotent-by-finite, a small step in the general direction of proving this for NIP ω-categorical groups. We also show that in dp-minimal definably amenable groups, f-generic global types are strongly f-generic.

Witnessing Dp-Rank

We prove that in NTP_2 theories if p is a dependent type with dp-rank >= \kappa, then this can be witnessed by indiscernible sequences of tuples satisfying p. If p has dp-rank infinity, then this can be witnessed by singletons (in any theory).

An embedding theorem of &#x1d53c;0 with model theoretic applications

We provide a new criterion for embedding 𝔼0, and apply it to equivalence relations in model theory. This generalize the results of the authors and Pierre Simon on the Borel cardinality of Lascar strong types equality, and Newelskis results about pseudo Fσ groups.

Exact saturation in simple and NIP theories

A theory

The Univalence Axiom in posetal model categories.

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