Itay Kaplan

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Forking and Dividing in NTP2 theories

2012-01-20

Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in … Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2 .

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Additivity of the dp-rank

2013-07-01

Itay Kaplan , Alf Onshuus , Alexander Usvyatsov

The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its … The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.

Artin-Schreier extensions in NIP and simple fields

2011-09-29

Itay Kaplan , Thomas Scanlon , Frank Olaf Wagner

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Groups and fields with $\operatorname {NTP}_{2}$

2014-08-19

Artem Chernikov , Itay Kaplan , Pierre Simon

$\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and … $\operatorname {NTP}_{2}$ is a large class of first-order theories defined by Shelah generalizing simple and NIP theories. Algebraic examples of $\operatorname {NTP}_{2}$ structures are given by ultra-products of $p$-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in $\operatorname {NTP}_{2}$ structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any $\operatorname {NTP}_{2}$ field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight) and show that every strongly dependent valued field is Kaplansky.

Chain conditions in dependent groups

2013-07-02

Itay Kaplan , Saharon Shelah

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The affine and projective groups are maximal

2015-11-12

Itay Kaplan , Pierre Simon

We show that the groups $AGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 2$) and $PGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 3$), seen as closed subgroups of $S_{\omega }$, are maximal-closed. We show that the groups $AGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 2$) and $PGL_{n}\left (\mathbb {Q}\right )$ (for $n\geq 3$), seen as closed subgroups of $S_{\omega }$, are maximal-closed.

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Local character of Kim-independence

2018-08-08

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

We show that $\mathrm {NSOP}_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is $\mathrm {NSOP}_{1}$, $M\models … We show that $\mathrm {NSOP}_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is $\mathrm {NSOP}_{1}$, $M\models T$, and $p$ is a complete type over $M$, then the collection of elementary substructures of size $\left |T\right |$ over which $p$ does not Kim-fork is a club of $\left [M\right ]^{\left |T\right |}$ and that this characterizes $\mathrm {NSOP}_{1}$. We also present a new phenomenon we call dual local-character for Kim-independence in $\mathrm {NSOP}_{1}$ theories.

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The Borel cardinality of Lascar strong types

2014-08-20

Itay Kaplan , Benjamin D. Miller , Pierre Simon

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 … 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).

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Transitivity of Kim-independence

2021-01-13

Itay Kaplan , Nicholas Ramsey

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EXAMPLES IN DEPENDENT THEORIES

2014-06-01

Itay Kaplan , Saharon Shelah

Abstract In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In … Abstract In the first part we show a counterexample to a conjecture by Shelah regarding the existence of indiscernible sequences in dependent theories (up to the first inaccessible cardinal). In the second part we discuss generic pairs, and give an example where the pair is not dependent. Then we define the notion of directionality which deals with counting the number of coheirs of a type and we give examples of the different possibilities. Then we discuss nonsplintering, an interesting notion that appears in the work of Rami Grossberg, Andrés Villaveces and Monica VanDieren, and we show that it is not trivial (in the sense that it can be different than splitting) whenever the directionality of the theory is not small. In the appendix we study dense types in RCF.

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DECIDABILITY AND CLASSIFICATION OF THE THEORY OF INTEGERS WITH PRIMES

2017-09-01

Itay Kaplan , Saharon Shelah

Abstract We show that under Dickson’s conjecture about the distribution of primes in the natural numbers, the theory Th (ℤ , +, 1, 0, Pr ) where Pr is a … Abstract We show that under Dickson’s 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.

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Automorphism groups of finite topological rank

2019-01-23

Itay Kaplan , Pierre Simon

We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite … We offer a criterion for showing that the automorphism group of an ultrahomogeneous structure is topologically 2-generated and even has a cyclically dense conjugacy class. We then show how finite topological rank of the automorphism group of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical structure can go down to reducts. Together, those results prove that a large number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical structures that appear in the literature have an automorphism group of finite topological rank. In fact, we are not aware of any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-categorical structure to which they do not apply (assuming the automorphism group has no compact quotients). We end with a few questions and conjectures.

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Witnessing Dp-Rank

2014-01-01

Itay Kaplan , Pierre Simon

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 … 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).

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Some Remarks on dp-Minimal Groups

2017-01-01

Itay Kaplan , Elad Levi , Pierre Simon

Exact saturation in simple and NIP theories

2017-02-02

Itay Kaplan , Saharon Shelah , Pierre Simon

A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: … A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

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A dependent theory with few indiscernibles

2014-06-20

Itay Kaplan , Saharon Shelah

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On non-forking spectra

2012-05-14

Artem Chernikov , Itay Kaplan , Saharon Shelah

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 … 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.

On non-forking spectra

2016-11-14

Artem Chernikov , Itay Kaplan , Saharon Shelah

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 … 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} .

SATURATED MODELS FOR THE WORKING MODEL THEORIST

2023-02-17

Yatir Halevi , Itay Kaplan

Abstract We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation … Abstract We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

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Decidability and classification of the theory of integers with primes

2016-01-26

Itay Kaplan , Saharon Shelah

We show that under Dickson's 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 … We show that under Dickson's 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.

Strict independence in dependent theories

2012-08-20

Itay Kaplan , Alexander Usvyatsov

We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular it follows from our investigation that in resilient theories … We investigate the notions of strict independence and strict non-forking, and establish basic properties and connections between the two. In particular it follows from our investigation that in resilient theories strict non-forking is symmetric. Based on this study, we develop notions of weight which characterize NTP2, dependence and strong dependence. Many of our proofs rely on careful analysis of sequences that witness dividing. We prove simple characterizations of such sequences in resilient theories, as well as of Morley sequences which are witnesses. As a by-product we obtain information on types co-dominated by generically stable types in dependent theories. For example, we prove that every Morley sequence in such a type is a witness.

Additivity of the dp-rank

2011-09-07

Itay Kaplan , Alf Onshuus , Alexander Usvyatsov

The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of … The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density.

The affine and projective groups are maximal

2013-10-30

Itay Kaplan , Pierre Simon

We show that the groups AGL_n(Q) and PGL_n(Q), seen as closed subgroups of S_{\infty}, are maximal-closed. We show that the groups AGL_n(Q) and PGL_n(Q), seen as closed subgroups of S_{\infty}, are maximal-closed.

Non-forking and preservation of NIP and dp-rank

2021-02-05

Pedro Andrés Estevan , Itay Kaplan

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An embedding theorem of 𝔼0 with model theoretic applications

2014-12-01

Itay Kaplan , Benjamin D. Miller

We provide a new criterion for embedding 𝔼 0 , and apply it to equivalence relations in model theory. This generalize the results of the authors and Pierre Simon on … We provide a new criterion for embedding 𝔼 0 , 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 Newelski's results about pseudo F σ groups.

On Kim-Independence

2017-02-13

Itay Kaplan , Nicholas Ramsey

We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of … We study NSOP$_{1}$ theories. We define Kim-independence, which generalizes non-forking independence in simple theories and corresponds to non-forking at a generic scale. We show that Kim-independence satisfies a version of Kim's lemma, local character, symmetry, and an independence theorem and that, moreover, these properties individually characterize NSOP$_{1}$ theories. We describe Kim-independence in several concrete theories and observe that it corresponds to previously studied notions of independence in Frobenius fields and vector spaces with a generic bilinear form.

Transitivity of Kim-independence

2019-01-21

Itay Kaplan , Nicholas Ramsey

We prove several results on the behavior of Kim-independence upon changing the base in NSOP$_{1}$ theories. As a consequence, we prove that Kim-independence satisfies transitivity and that this characterizes NSOP$_{1}$. … We prove several results on the behavior of Kim-independence upon changing the base in NSOP$_{1}$ theories. As a consequence, we prove that Kim-independence satisfies transitivity and that this characterizes NSOP$_{1}$. Moreover, we characterize witnesses to Kim-dividing as exactly the $\ind^{K}$-Morley sequences. We give several applications, answering a number of open questions concerning transitivity, Morley sequences, and local character in NSOP$_{1}$ theories.

On uniform definability of types over finite sets for NIP formulas

2020-10-15

Shlomo Eshel , Itay Kaplan

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles … Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets (UDTFS). This settles a conjecture of Laskowski.

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ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE

2021-02-01

Itay Kaplan , Tomasz Rzepecki , Daoud Siniora

Abstract We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms. Abstract We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

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Existentially closed models of fields with a distinguished submodule

2021-01-01

Christian d’Elbée , Itay Kaplan , Leor Neuhauser

This paper deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and … This paper deals with the class of existentially closed models of fields with a distinguished submodule (over a fixed subring). In the positive characteristic case, this class is elementary and was investigated by the first-named author. Here we study this class in Robinson's logic, meaning the category of existentially closed models with embeddings following Haykazyan and Kirby, and prove that in this context this class is NSOP$_1$ and TP$_2$.

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Non-forking and preservation of NIP and dp-rank

2019-01-01

Pedro Andrés Estevan , Itay Kaplan

We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question … We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question for dp-rank. We show that if $B$ contains a Morley sequence $I$ generated by $p$ over $A$, then $p\restriction AI$ is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of "trees whose open cones are models of some theory" and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.

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Automorphism towers and automorphism groups of fields without Choice

2012-01-01

Itay Kaplan , Saharon Shelah

This paper can be viewed as a continuation of [KS09] that dealt with the automorphism tower problem without Choice. Here we deal with the inequation which connects the automorphism tower … This paper can be viewed as a continuation of [KS09] that dealt with the automorphism tower problem without Choice. Here we deal with the inequation which connects the automorphism tower and the normalizer tower without Choice and introduce a new proof to a theorem of Fried and Kollar that any group can be represented as an automorphism group of a field. The proof uses a simple construction: working more in graph theory, and less in algebra.

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The automorphism tower of a centerless group without Choice

2009-10-05

Itay Kaplan , Saharon Shelah

Forking in NTP_2 theories

2009-06-15

Artem Chernikov , Itay Kaplan

We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact … We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP_2.

Local character of Kim-independence

2017-07-10

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and … We show that NSOP$_{1}$ theories are exactly the theories in which Kim-independence satisfies a form of local character. In particular, we show that if $T$ is NSOP$_{1}$, $M\models T$, and $p$ is a type over $M$, then the collection of elementary substructures of size $\left|T\right|$ over which $p$ does not Kim-fork is a club of $\left[M\right]^{\left|T\right|}$ and that this characterizes NSOP$_{1}$. We also present a new phenomenon we call dual local-character for Kim-independence in NSOP$_{1}$-theories.

Automorphism groups of finite topological rank

2017-09-06

Itay Kaplan , Pierre Simon

Forcing a countable structure to belong to the ground model

2016-12-01

Itay Kaplan , Saharon Shelah

Suppose that P is a forcing notion, L is a language (in ), a P-name such that " is a countable L-structure". In the product , there are names such … Suppose that P is a forcing notion, L is a language (in ), a P-name such that " is a countable L-structure". In the product , there are names such that for any generic filter over , and . Zapletal asked whether or not implies that there is some such that . We answer this question negatively and discuss related issues.

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On algebraically closed fields with a distinguished subfield

2024-04-24

Christian d’Elbée , Itay Kaplan , Leor Neuhauser

An embedding theorem of $\mathbb{E}_{0}$ with model theoretic applications

2013-01-01

Itay Kaplan , Benjamin D. Miller

We provide a new criterion for embedding $\mathbb{E}_{0}$, and apply it to equivalence relations in model theory. This generalize the results of the authors and Pierre Simon on the Borel … We provide a new criterion for embedding $\mathbb{E}_{0}$, 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 Newelski's results about pseudo $F_σ$ groups.

Groups and fields with NTP2

2012-12-26

Artem Chernikov , Itay Kaplan , Pierre Simon

NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain … NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain valued difference fields (such as a non-standard Frobenius automorphism living on an algebraically closed valued field of characteristic 0). In this note we present some results on groups and fields definable in NTP2 structures. Most importantly, we isolate a chain condition for definable normal subgroups and use it to show that any NTP2 field has only finitely many Artin-Schreier extensions. We also discuss a stronger chain condition coming from imposing bounds on burden of the theory (an appropriate analogue of weight), and show that every strongly dependent valued field is Kaplansky.

Representing groups as automorphism groups of fields without Choice

2010-04-11

Itay Kaplan , Saharon Shelah

The generic pair conjecture for dependent finite diagrams

2016-01-07

Itay Kaplan , Noa Lavi , Saharon Shelah

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Some NIP-like phenomena in NTP$_{2}$

2017-01-01

Itay Kaplan , Pierre Simon

We introduce the notion of an NTP$_{2}$-smooth measure and prove that they exist assuming NTP$_{2}$. Using this, we propose a notion of distality in NTP$_{2}$ that unfortunately does not intersect … We introduce the notion of an NTP$_{2}$-smooth measure and prove that they exist assuming NTP$_{2}$. Using this, we propose a notion of distality in NTP$_{2}$ that unfortunately does not intersect simple theories trivially. We then prove a finite alternation theorem for a subclass of NTP$_{2}$ that contains resilient theories. In the last section we prove that under NIP, any type over a model of singular size is finitely satisfiable in a smaller model, and ask if a parallel result (with non-forking replacing finite satisfiability) holds in NTP$_{2}$.

On the automorphism group of the universal homogeneous meet-tree

2019-04-10

Itay Kaplan , Tomasz Rzepecki , Daoud Siniora

We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms. We show that the countable universal homogeneous meet-tree has a generic automorphism, but it does not have a generic pair of automorphisms.

Chain conditions in dependent groups

2011-12-04

Itay Kaplan , Saharon Shelah

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. 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.

A generalisation of von Staudt’s theorem on cross-ratios

2019-02-22

Yatir Halevi , Itay Kaplan

Abstract A generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that … Abstract A generalisation of von Staudt’s theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

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Criteria for exact saturation and singular compactness

2020-03-01

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. … We introduce the class of unshreddable theories, which contains the simple and NIP theories, and prove that such theories have exactly saturated models in singular cardinals, satisfying certain set-theoretic hypotheses. We also give criteria for a theory to have singular compactness.

Boolean Types in Dependent Theories

2020-08-07

Itay Kaplan , Ori Segel , Saharon Shelah

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some … The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra B to each formula. We show some basic results regarding the effect of the properties of B on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author's result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures - in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

Criteria for exact saturation and singular compactness

2021-05-13

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

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Existentially closed models of fields with a distinguished submodule

2021-10-05

Christian d’Elbée , Itay Kaplan , Leor Neuhauser

Automorphisms of the Rado meet-tree

2025-03-01

Itay Kaplan , Binyamin Riahi , Arturo Rodríguez Fanlo

Stable reducts of elementary extensions of Presburger arithmetic

2024-12-13

Eran Alouf , Antongiulio Fornasiero , Itay Kaplan

Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < \omega$, $[G: nG]$ is finite). Then every stable reduct … Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < \omega$, $[G: nG]$ is finite). Then every stable reduct of $N$ which expands $(G,+)$ (equivalently every reduct that does not add new unary definable sets) is interdefinable with $(G,+)$. This extends previous results on stable reducts of $(\mathbb{Z}, +, <)$ to (stable) reducts of elementary extensions of $\mathbb{Z}$. In particular this holds for $G = \mathbb{Z}$ and $G = \mathbb{Q}$. As a result we answer a question of Conant from 2018.

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Infinite Cliques in Simple and Stable Graphs

2024-08-10

Yatir Halevi , Itay Kaplan , Saharon Shelah

Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. … Suppose that $G$ is a graph of cardinality $\mu^+$ with chromatic number $\chi(G)\geq \mu^+$. One possible reason that this could happen is if $G$ contains a clique of size $\mu^+$. We prove that this is indeed the case when the edge relation is stable. When $G$ is a random graph (which is simple but not stable), this is not true. But still if in general the complete theory of $G$ is simple, $G$ must contain finite cliques of unbounded sizes.

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On algebraically closed fields with a distinguished subfield

2024-04-24

Christian d’Elbée , Itay Kaplan , Leor Neuhauser

Density of compressible types and some consequences

2024-02-18

Martin Bays , Itay Kaplan , Pierre Simon

We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), … We study compressible types in the context of (local and global) NIP. By extending a result in machine learning theory (the existence of a bound on the recursive teaching dimension), we prove density of compressible types. Using this, we obtain explicit uniform honest definitions for NIP formulas (answering a question of Eshel and the second author), and build compressible models in countable NIP theories.

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Generic Stability Independence and Treeless Theories

2024-01-01

Itay Kaplan , Nicholas Ramsey , Pierre Simon

Abstract We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that … Abstract We initiate a systematic study of generic stability independence and introduce the class of treeless theories in which this notion of independence is particularly well behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP $_{3}$ theory is simple.

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ON LARGE EXTERNALLY DEFINABLE SETS IN NIP

2023-12-04

Martin Bays , Omer Ben-Neria , Itay Kaplan +1 more

Abstract We study cofinal systems of finite subsets of $\omega _1$ . We show that while such systems can be NIP, they cannot be defined in an NIP structure. We … Abstract We study cofinal systems of finite subsets of $\omega _1$ . We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: In an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.

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A definable (p,q)-theorem for NIP theories

2023-11-17

Itay Kaplan

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Infinite stable graphs with large chromatic number II

2023-06-14

Yatir Halevi , Itay Kaplan , Saharon Shelah

We prove a version of the strong Taylor’s conjecture for stable graphs: if G is a stable graph whose chromatic number is strictly greater than \beth_2(\aleph_0) then G contains all … We prove a version of the strong Taylor’s conjecture for stable graphs: if G is a stable graph whose chromatic number is strictly greater than \beth_2(\aleph_0) then G contains all finite subgraphs of \textup{Sh}_n(\omega) and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model-theoretic ingredient is a generalization of the classical construction of Ehrenfeucht–Mostowski models to an infinitary setting, giving a new characterization of stability.

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SATURATED MODELS FOR THE WORKING MODEL THEORIST

2023-02-17

Yatir Halevi , Itay Kaplan

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The model theory of geometric random graphs

2023-01-01

Omer Ben-Neria , Itay Kaplan , Tingxiang Zou

We study the logical properties of infinite geometric random graphs, introduced by Bonato and Janssen. These are graphs whose vertex set is a dense ``generic'' subset of a metric space, … We study the logical properties of infinite geometric random graphs, introduced by Bonato and Janssen. These are graphs whose vertex set is a dense ``generic'' subset of a metric space, where two vertices are adjacent with probability $p>0$ provided the distance between them is bounded by some constant number. We prove that for a large class of metric spaces, including circles, spheres and the complete Urysohn space, almost all geometric random graphs on a given space are elementary equivalent. Moreover, their first-order theory can reveal geometric properties of the underlying metric space.

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Generic Stability Independence and Treeless theories

2023-01-01

Itay Kaplan , Nicholas Ramsey , Pierre Simon

We initiate a systematic study of \emph{generic stability independence} and introduce the class of \emph{treeless theories} in which this notion of independence is particularly well-behaved. We show that the class … We initiate a systematic study of \emph{generic stability independence} and introduce the class of \emph{treeless theories} in which this notion of independence is particularly well-behaved. We show that the class of treeless theories contains both binary theories and stable theories and give several applications of the theory of independence for treeless theories. As a corollary, we show that every binary NSOP$_{3}$ theory is simple.

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Automorphisms of the Rado meet-tree

2023-01-01

Itay Kaplan , Binyamin Riahi , Arturo Rodriguez Fanlo

We prove that the group of automorphisms of the generic meet-tree expansion of an infinite non-unary free Fra\"{\i}ss\'{e} limit over a finite relational language is simple. As a prototypical case, … We prove that the group of automorphisms of the generic meet-tree expansion of an infinite non-unary free Fra\"{\i}ss\'{e} limit over a finite relational language is simple. As a prototypical case, the group of automorphism of the Rado meet-tree (i.e. the Fra\"{\i}ss\'{e} limit of finite graphs which are also meet-trees) is simple.

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Exact saturation in pseudo-elementary classes for simple and stable theories

2022-06-17

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study … We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.

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BOOLEAN TYPES IN DEPENDENT THEORIES

2022-01-26

Itay Kaplan , Ori Segel , Saharon Shelah

Abstract The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We … Abstract The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

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On large externally definable sets in NIP

2022-01-01

Martin Bays , Omer Ben-Neria , Itay Kaplan +1 more

We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive … We study cofinal systems of finite subsets of $\omega_1$. We show that while such systems can be NIP, they cannot be defined in an NIP structure. We deduce a positive answer to a question of Chernikov and Simon from 2013: in an NIP theory, any uncountable externally definable set contains an infinite definable subset. A similar result holds for larger cardinals.

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A definable $(p,q)$-theorem for NIP theories

2022-01-01

Itay Kaplan

We prove a definable version of Matou\v{s}ek's $(p,q)$-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version. The proof builds on a … We prove a definable version of Matou\v{s}ek's $(p,q)$-theorem in NIP theories. This answers a question of Chernikov and Simon. We also prove a uniform version. The proof builds on a proof of Boxall and Kestner who proved this theorem in the distal case, utilizing the notion of locally compressible types which appeared in the work of the author with Bays and Simon.

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Infinite stable graphs with large chromatic number

2021-12-08

Yatir Halevi , Itay Kaplan , Saharon Shelah

We prove that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals left-parenthesis upper V comma upper E right-parenthesis"><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">G=(V,E)</mml:annotation></mml:semantics></mml:math></inline-formula>is an<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"><mml:semantics><mml:mi>ω</mml:mi><mml:annotation encoding="application/x-tex">\omega</mml:annotation></mml:semantics></mml:math></inline-formula>-stable (respectively, superstable) graph with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" … We prove that if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals left-parenthesis upper V comma upper E right-parenthesis"><mml:semantics><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>V</mml:mi><mml:mo>,</mml:mo><mml:mi>E</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">G=(V,E)</mml:annotation></mml:semantics></mml:math></inline-formula>is an<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"><mml:semantics><mml:mi>ω</mml:mi><mml:annotation encoding="application/x-tex">\omega</mml:annotation></mml:semantics></mml:math></inline-formula>-stable (respectively, superstable) graph with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="chi left-parenthesis upper G right-parenthesis greater-than normal alef 0"><mml:semantics><mml:mrow><mml:mi>χ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">\chi (G)>\aleph _0</mml:annotation></mml:semantics></mml:math></inline-formula>(respectively,<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript normal alef 0"><mml:semantics><mml:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi mathvariant="normal">ℵ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:msup><mml:annotation encoding="application/x-tex">2^{\aleph _0}</mml:annotation></mml:semantics></mml:math></inline-formula>) then<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>contains all the finite subgraphs of the shift graph<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal h Subscript n Baseline left-parenthesis omega right-parenthesis"><mml:semantics><mml:mrow><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi mathvariant="normal">S</mml:mi><mml:mi mathvariant="normal">h</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>ω</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">\mathrm {Sh}_n(\omega )</mml:annotation></mml:semantics></mml:math></inline-formula>for some<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"><mml:semantics><mml:mi>n</mml:mi><mml:annotation encoding="application/x-tex">n</mml:annotation></mml:semantics></mml:math></inline-formula>. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"><mml:semantics><mml:mi>G</mml:mi><mml:annotation encoding="application/x-tex">G</mml:annotation></mml:semantics></mml:math></inline-formula>is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"><mml:semantics><mml:mi>ω</mml:mi><mml:annotation encoding="application/x-tex">\omega</mml:annotation></mml:semantics></mml:math></inline-formula>-stable with<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper U left-parenthesis upper G right-parenthesis less-than-or-equal-to 2"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">U</mml:mi><mml:mo>⁡</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">\operatorname {U}(G)\leq 2</mml:annotation></mml:semantics></mml:math></inline-formula>we prove that<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n less-than-or-equal-to 2"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>≤</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:annotation encoding="application/x-tex">n\leq 2</mml:annotation></mml:semantics></mml:math></inline-formula>suffices.

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Saturated Models for the Working Model Theorist

2021-12-05

Yatir Halevi , Itay Kaplan

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is … We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

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Existentially closed models of fields with a distinguished submodule

2021-10-05

Christian d’Elbée , Itay Kaplan , Leor Neuhauser

Criteria for exact saturation and singular compactness

2021-05-13

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

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Non-forking and preservation of NIP and dp-rank

2021-02-05

Pedro Andrés Estevan , Itay Kaplan

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ON THE AUTOMORPHISM GROUP OF THE UNIVERSAL HOMOGENEOUS MEET-TREE

2021-02-01

Itay Kaplan , Tomasz Rzepecki , Daoud Siniora

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Transitivity of Kim-independence

2021-01-13

Itay Kaplan , Nicholas Ramsey

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Existentially closed models of fields with a distinguished submodule

2021-01-01

Christian d’Elbée , Itay Kaplan , Leor Neuhauser

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Infinite Stable Graphs With Large Chromatic Number II

2021-01-01

Yatir Halevi , Itay Kaplan , Saharon Shelah

We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all … We prove a version of the strong Taylor's conjecture for stable graphs: if $G$ is a stable graph whose chromatic number is strictly greater than $\beth_2(\aleph_0)$ then $G$ contains all finite subgraphs of Sh$_n(\omega)$ and thus has elementary extensions of unbounded chromatic number. This completes the picture from our previous work. The main new model theoretic ingredient is a generalization of the classical construction of Ehrenfeucht-Mostowski models to an infinitary setting, giving a new characterization of stability.

On algebraically closed fields with a distinguished subfield

2021-01-01

Christian d’Elbée , Itay Kaplan , Leor Neuhauser

This paper is concerned with the model-theoretic study of pairs $(K,F)$ where $K$ is an algebraically closed field and $F$ is a distinguished subfield of $K$ allowing extra structure. We … This paper is concerned with the model-theoretic study of pairs $(K,F)$ where $K$ is an algebraically closed field and $F$ is a distinguished subfield of $K$ allowing extra structure. We study the basic model-theoretic properties of those pairs, such as quantifier elimination, model-completeness and saturated models. We also prove some preservation results of classification-theoretic notions such as stability, simplicity, NSOP$_1$, and NIP. As an application, we conclude that a PAC field is NSOP$_1$ iff its absolute Galois group is (as a profinite group).

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Density of compressible types and some consequences

2021-01-01

Martin Bays , Itay Kaplan , Pierre Simon

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Saturated Models for the Working Model Theorist

2021-01-01

Yatir Halevi , Itay Kaplan

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On uniform definability of types over finite sets for NIP formulas

2020-10-15

Shlomo Eshel , Itay Kaplan

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Boolean Types in Dependent Theories

2020-08-07

Itay Kaplan , Ori Segel , Saharon Shelah

Infinite Stable Graphs With Large Chromatic Number

2020-07-23

Yatir Halevi , Itay Kaplan , Saharon Shelah

We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(\omega)$ for some $n$. … We prove that if $G=(V,E)$ is an $\omega$-stable (respectively, superstable) graph with $\chi(G)>\aleph_0$ (respectively, $2^{\aleph_0}$) then $G$ contains all the finite subgraphs of the shift graph $\text{Sh}_n(\omega)$ for some $n$. We prove a variant of this theorem for graphs interpretable in stationary stable theories. Furthermore, if $G$ is $\omega$-stable with $\mathrm{U}(G)\leq 2$ we prove that $n\leq 2$ suffices.

Criteria for exact saturation and singular compactness

2020-03-01

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

Exact saturation in pseudo-elementary classes for simple and stable theories

2020-01-01

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable … We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.

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Boolean Types in Dependent Theories

2020-01-01

Itay Kaplan , Ori Segel , Saharon Shelah

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Criteria for exact saturation and singular compactness

2020-01-01

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

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Infinite Stable Graphs With Large Chromatic Number

2020-01-01

Yatir Halevi , Itay Kaplan , Saharon Shelah

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Non-forking and preservation of NIP and dp-rank

2019-09-10

Pedro Andrés Estevan , Itay Kaplan

We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question … We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question for dp-rank. We show that if $B$ contains a Morley sequence $I$ generated by $p$ over $A$, then $p\restriction AI$ is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of whose open cones are models of some and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.

On uniform definability of types over finite sets for NIP formulas

2019-04-23

Shlomo Eshel , Itay Kaplan

On the automorphism group of the universal homogeneous meet-tree

2019-04-10

Itay Kaplan , Tomasz Rzepecki , Daoud Siniora

A generalisation of von Staudt’s theorem on cross-ratios

2019-02-22

Yatir Halevi , Itay Kaplan

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Automorphism groups of finite topological rank

2019-01-23

Itay Kaplan , Pierre Simon

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Transitivity of Kim-independence

2019-01-21

Itay Kaplan , Nicholas Ramsey

Non-forking and preservation of NIP and dp-rank

2019-01-01

Pedro Andrés Estevan , Itay Kaplan

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On uniform definability of types over finite sets for NIP formulas

2019-01-01

Shlomo Eshel , Itay Kaplan

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Transitivity of Kim-independence

2019-01-01

Itay Kaplan , Nicholas Ramsey

Local character of Kim-independence

2018-08-08

Itay Kaplan , Nicholas Ramsey , Saharon Shelah

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A Generalization of von Staudt's Theorem on Cross-Ratios

2018-03-22

Yatir Halevi , Itay Kaplan

A generalization of von Staudt's theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any … A generalization of von Staudt's theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the projective semilinear group over an algebraically closed field of transcendence degree at least 1 is 4-transitive.

A Generalization of von Staudt's Theorem on Cross-Ratios

2018-01-01

Yatir Halevi , Itay Kaplan

Automorphism groups of finite topological rank

2017-09-06

Itay Kaplan , Pierre Simon

Coauthor	Papers Together
Saharon Shelah	49
Pierre Simon	24
Nicholas Ramsey	15
Artem Chernikov	12
Yatir Halevi	11
Alexander Usvyatsov	6
Leor Neuhauser	4
Christian d’Elbée	4
Martin Bays	4
Elad Levi	3
Misha Gavrilovich	3
Assaf Hasson	3
Shlomo Eshel	3
Alf Onshuus	3
Omer Ben-Neria	3
Benjamin D. Miller	3
Ori Segel	3
Pedro Andrés Estevan	3
Thomas Scanlon	2
Tomasz Rzepecki	2
Frank Olaf Wagner	2
Binyamin Riahi	2
Daoud Siniora	2
Noa Lavi	2
Eran Alouf	1
Arturo Rodríguez Fanlo	1
Tingxiang Zou	1
Antongiulio Fornasiero	1
Arturo Rodriguez Fanlo	1
Noa Lavi	1
Yatir Halevi	1

Dependent first order theories, continued

2009-09-01

Saharon Shelah

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On dp-minimal ordered structures

2011-05-19

Pierre Simon

Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and … Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.

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On dp-minimality, strong dependence and weight

2011-07-06

Alf Onshuus , Alexander Usvyatsov

Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight. Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.

On model-theoretic tree properties

2016-11-22

Artem Chernikov , Nicholas Ramsey

We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see … We study model theoretic tree properties ([Formula: see text]) and their associated cardinal invariants ([Formula: see text], respectively). In particular, we obtain a quantitative refinement of Shelah’s theorem ([Formula: see text]) for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable (partially answering a question of Shelah) and that weak [Formula: see text] is equivalent to [Formula: see text] (answering a question of Kim and Kim). Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are indeed [Formula: see text].

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Theories without the tree property of the second kind

2013-10-28

Artem Chernikov

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Forking and Dividing in NTP2 theories

2012-01-20

Artem Chernikov , Itay Kaplan

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On NIP and invariant measures

2011-05-13

Ehud Hrushovski , Anand Pillay

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.

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Distal and non-distal NIP theories

2012-11-01

Pierre Simon

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Additivity of the dp-rank

2013-07-01

Itay Kaplan , Alf Onshuus , Alexander Usvyatsov

Externally definable sets and dependent pairs II

2015-02-20

Artem Chernikov , Pierre Simon

We continue investigating the structure of externally definable sets in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper N normal upper I normal upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mi mathvariant="normal">I</mml:mi> … We continue investigating the structure of externally definable sets in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper N normal upper I normal upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mi mathvariant="normal">I</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {NIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> theories and preservation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper N normal upper I normal upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mi mathvariant="normal">I</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {NIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper N normal upper I normal upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mi mathvariant="normal">I</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {NIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, while naming a large one doesn’t; there are models of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper N normal upper I normal upper P"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">N</mml:mi> <mml:mi mathvariant="normal">I</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {NIP}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> theories over which all 1-types are definable, but not all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-types.

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Studies in Logic and the Foundations of Mathematics

1990-01-01

On generically stable types in dependent theories

2009-01-04

Alexander Usvyatsov

Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories. Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.

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EXAMPLES IN DEPENDENT THEORIES

2014-06-01

Itay Kaplan , Saharon Shelah

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Witnessing Dp-Rank

2014-01-01

Itay Kaplan , Pierre Simon

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Strongly dependent theories

2014-07-28

Saharon Shelah

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Thorn orthogonality and domination in unstable theories

2011-01-01

Alf Onshuus , Alexander Usvyatsov

We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories. We study orthogonality, domination, weight, regular and minimal types in the contexts of rosy and super-rosy theories.

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Dependent dreams: recounting types

2012-01-01

Saharon Shelah

We investigate the class of models of a general dependent theory. We continue math.LO/0702292 in particular investigating so called "decomposition of types"; thesis is that what holds for stable theory … We investigate the class of models of a general dependent theory. We continue math.LO/0702292 in particular investigating so called "decomposition of types"; thesis is that what holds for stable theory and for Th(Q, 2^{|T|} then is indiscernible for some stationary S subseteq kappa. Third, for stable T,a model is kappa-saturated iff it is aleph_epsilon-saturated and every infinite indiscernible set (of elements) of cardinality < kappa can be increased. We prove here an analog. Fourth, for p in S(M), the number of ultrafilters on the outside definable subsets of M extending p has an absolute bound 2^{|T|} . Restricting ourselves to one phi(x, y), the number is finite, with an absolute found (well depending on T and phi).

Dependent theories and the generic pair conjecture

2014-10-20

Saharon Shelah

We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis … We try to understand complete types over a somewhat saturated model of a complete first-order theory which is dependent (previously called NIP), by "decomposition theorems for such types". Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first, we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular, we try to prove the generic pair conjecture and do it for measurable cardinals.

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The automorphism tower problem II

1998-12-01

Simon Thomas

Stable theories and representation over sets

2016-05-01

Saharon Shelah , M. Cohen

In this paper we give characterizations of the stable and ℵ 0 ‐stable theories, in terms of an external property called representation. In the sense of the representation property, the … In this paper we give characterizations of the stable and ℵ 0 ‐stable theories, in terms of an external property called representation. In the sense of the representation property, the mentioned classes of first‐order theories can be regarded as “not very complicated”.

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Automorphism Groups of Infinite Semilinear Orders (I)

1989-05-01

Manfred Droste , W. Charles Holland , Dugald Macpherson

Results are obtained concerning normal subgroups of the automorphism groups of certain infinite trees. These structures are mostly ℵo-categorical, and are trees in a poset-theoretic but not graph-theoretic sense. It … Results are obtained concerning normal subgroups of the automorphism groups of certain infinite trees. These structures are mostly ℵo-categorical, and are trees in a poset-theoretic but not graph-theoretic sense. It is shown that the automorphism group has a smallest non-trivial normal subgroup, a largest proper normal subgroup, and at least 22ο normal subgroups between these two. We also obtain and use some results on groups of automorphisms of chains.

CLASSIFICATION THEORY FOR ELEMENTARY CLASSES WITH THE DEPENDENCE PROPERTY-A MODEST BEGINNING

2004-03-01

Saharon Shelah

Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the … Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the independence property) there is a substantial theory which means: a substantial body of basic results for all such classes and some complimentary results for the first order theories with the independence property, as for the family of stable (and the family of simple) first order theories. We examine some properties.

Turbulence, amalgamation, and generic automorphisms of homogeneous structures

2006-11-27

Alexander S. Kechris , Christian Rosendal

We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological … We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many ω-stable, ℵ0-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S∞ has a unique non-trivial separable group topology. For several interesting groups we also establish Serre's properties (FH) and (FA).

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A dependent theory with few indiscernibles

2014-06-20

Itay Kaplan , Saharon Shelah

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DP-MINIMALITY: INVARIANT TYPES AND DP-RANK

2014-12-01

Pierre Simon

Abstract This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version … Abstract This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality. In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimension coming from the order.

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On uniform definability of types over finite sets

2012-04-04

Vincent Guingona

Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called “uniform definability of types over finite sets” … Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called “uniform definability of types over finite sets” (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

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Minimal bounded index subgroup for dependent theories

2007-11-09

Saharon Shelah

For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> … For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathfrak {C}}_{T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every type definable group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.

A survey of homogeneous structures

2011-02-22

Dugald Macpherson

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Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups

2005-02-01

Alexander S. Kechris , Vladimir Pestov , Stevo Todorčević

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Exact saturation in simple and NIP theories

2017-02-02

Itay Kaplan , Saharon Shelah , Pierre Simon

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Type decomposition in NIP theories

2019-10-28

Pierre Simon

A first order theory is NIP if all definable families of subsets have finite VC-dimension. We provide a justification for the intuition that NIP structures should be a combination of … A first order theory is NIP if all definable families of subsets have finite VC-dimension. We provide a justification for the intuition that NIP structures should be a combination of stable and order-like components. More precisely, we prove that any type in an NIP theory can be decomposed into a stable part (a generically stable partial type) and an order-like quotient.

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Dp-Minimality: Basic Facts and Examples

2011-07-01

Alfred Dolich , John Goodrick , David Lippel

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of … We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.

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Logical stability in group theory

1976-05-01

John T. Baldwin , Jan Saxl

Abstract This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety … Abstract This paper investigates the logical stability of various groups. Theorem 1: If a group G is stable and locally nilpotent then it is solvable. Theorem 2: Every non-Abelian variety of groups is unstable.

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Explanation of Independence

2005-01-01

Hans H. Adler

An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is … An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is characterised in terms of modular pairs in the lattice of algebraically closed sets. Wherever possible, forking and thorn-forking are treated in a uniform way. They are dual in the sense that forking is the finest (most restrictive) and thorn-forking the coarsest independence relation worth examining. We finish by defining the kernel of a sequence of indiscernibles and studying its relation to canonical bases.

The 116 reducts of (ℚ, <, a)

2008-09-01

Markus Junker , Martin Ziegler

Abstract This article aims to classify those reducts of expansions of (ℚ, <) by unary predicates which eliminate quantifiers, and in particular to show that, up to interdefinability, there are … Abstract This article aims to classify those reducts of expansions of (ℚ, <) by unary predicates which eliminate quantifiers, and in particular to show that, up to interdefinability, there are only finitely many for a given language. Equivalently, we wish to classify the closed subgroups of Sym(ℚ) containing the group of all automorphisms of (ℚ, <) fixing setwise certain subsets. This goal is achieved for expansions by convex predicates, yielding expansions by constants as a special case, and for the expansion by a dense, co-dense predicate. Partial results are obtained in the general setting of several dense predicates.

Types dans les corps valués munis d'applications coefficients

1999-06-01

Luc Bélair

We transpose Delon's analysis of types in valued fields to unramified henselian valued fields of mixed characteristic, by using coefficient maps of order $n$. This yields the Ax-Kochen-Ershov transfer principle … We transpose Delon's analysis of types in valued fields to unramified henselian valued fields of mixed characteristic, by using coefficient maps of order $n$. This yields the Ax-Kochen-Ershov transfer principle for the independence property in this class of valued fields.

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The Automorphism Tower Problem Revisited

1999-12-01

Winfried Just , Saharon Shelah , Simon Thomas

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On theories without the tree property of the second kind

2012-10-08

Artem Chernikov

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 … 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.

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Some Remarks on dp-Minimal Groups

2017-01-01

Itay Kaplan , Elad Levi , Pierre Simon

Six classes of theories

1976-05-01

H. Jerome Keisler

A theory T is said to κ-stable if, given a pair of models U ⊂ B of T with U of power κ, there are only κ types of elements … A theory T is said to κ-stable if, given a pair of models U ⊂ B of T with U of power κ, there are only κ types of elements of B over U (types are defined below). This notion was introduced by Morley (1965) who gave a powerful analysis of ω-stable theories. Shelah (1971) showed that there are only four possibilities for the set of κ in which a countable theory is stable. This partition of all theories into four classes (ω-stable, superstable, stable, and unstable theories) has proved to be of great value. However, most familiar examples of theories are unstable.

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Externally definable sets and dependent pairs

2012-05-12

Artem Chernikov , Pierre Simon

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Dependent T and existence of limit models

2006-09-22

Saharon Shelah

Does the class of linear orders have (one of the variants of) the so called (lambda, kappa)-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we … Does the class of linear orders have (one of the variants of) the so called (lambda, kappa)-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get some positive, i.e. existence results. More generally, letting T be a complete first order theory and for simplicity assume G.C.H., for regular lambda > kappa > |T| does T have (variants of) a (lambda,kappa)-limit models, except for stable T? For some, yes, the theory of dense linear order, for some, no. Moreover, for independent T we get negative, i.e. non-existence results. We deal more with linear orders.

An independence theorem for NTP2 theories

2014-01-01

Itaï Ben Yaacov , Artem Chernikov

We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce … We establish several results regarding dividing and forking in NTP2 theories. We show that dividing is the same as array-dividing. Combining it with existence of strictly invariant sequences we deduce that forking satisfies the chain condition over extension bases (namely, the forking ideal is S1, in Hrushovski's terminology). Using it we prove an independence theorem over extension bases (which, in the case of simple theories, specializes to the ordinary independence theorem). As an application we show that Lascar strong type and compact strong type coincide over extension bases in an NTP2 theory. We also define the dividing order of a theory -- a generalization of Poizat's fundamental order from stable theories -- and give some equivalent characterizations under the assumption of NTP2. The last section is devoted to a refinement of the class of strong theories and its place in the classification hierarchy.

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Generic stability, regularity, and quasiminimality

2011-09-07

Anand Pillay , Predrag Tanović

We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly … We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly regular types (p(x), x = x). We prove that in a theory without the strict order property, regular types are generically stable, and prove analogous results for quasiminimal structures. We prove that the “generic type” of a quasiminimal structure is “locally strongly regular”.

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Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory

1971-10-01

Saharon Shelah

GENERIC STABILITY AND STABILITY

2014-03-01

Hans H. Adler , Enrique Casanovas , Anand Pillay

Abstract We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second … Abstract We prove two results about generically stable types p in arbitrary theories. The first, on existence of strong germs, generalizes results from [2] on stably dominated types. The second is an equivalence of forking and dividing, assuming generic stability of p ( m ) for all m . We use the latter result to answer in full generality a question posed by Hasson and Onshuus: If P ( x ) ε S ( B ) is stable and does not fork over A then prestrictionA is stable. (They had solved some special cases.)

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Toward classifying unstable theories

1996-08-01

Saharon Shelah

Almost-disjoint sets the dense set problem and the partition calculus

1976-05-01

James E. Baumgartner

Lascar strong types in some simple theories

1999-06-01

Steven Buechler

Abstract In this paper a class of simple theories, called the low theories is developed, and the following is proved. T heorem . Let T be a low theory, A … Abstract In this paper a class of simple theories, called the low theories is developed, and the following is proved. T heorem . Let T be a low theory, A a set and a, b elements realizing the same strong type over A. Then, a and b realize the same Lasear strong type over A .

Invariants Related to the Tree Property

2015-11-19

Nicholas Ramsey

We consider global analogues of model-theoretic tree properties. The main objects of study are the invariants related to Shelah's tree property $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, and $\kappa_{\text{inp}}(T)$ and the relations that obtain … We consider global analogues of model-theoretic tree properties. The main objects of study are the invariants related to Shelah's tree property $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, and $\kappa_{\text{inp}}(T)$ and the relations that obtain between them. From strong colorings, we construct theories $T$ with $\kappa_{\text{cdt}}(T) > \kappa_{\text{sct}}(T) + \kappa_{\text{inp}}(T)$. We show that these invariants have distinct structural consequences, by investigating the decay of saturation in ultrapowers of models of $T$, where $T$ is some theory with $\kappa_{\text{cdt}}(T)$, $\kappa_{\text{sct}}(T)$, or $\kappa_{\text{inp}}(T)$ large and bounded. This answers some questions of Shelah.