On n-Dependence

Chernikov, Artem; Palacín, Daniel; Takeuchi, Kota

doi:10.1215/00294527-2019-0002

Cited by 20 publications

(47 citation statements)

References 21 publications

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“…In Theorem 3.15, we generalize this to simply Ramseyexpandable classes. This partially captures a generalization of the standard collapse-of-indiscernible results found in the literature [2,4,17]. Theorem 3.14.…”

Section: Collapse Of Indiscerniblessupporting

confidence: 63%

“…Though Theorem 3.14 is certainly an interesting result in its own right, it does not match the flavor of collapse-of-indiscernible results discovered thus far (in, say, [2], [4], and [17]). "Natural" dividing-lines typically do not involve linear orders.…”

Section: Collapse Of Indiscerniblesmentioning

confidence: 81%

“…One thing missing from the discussion here is precisely how these indiscernibles resist collapse. The results from [2], [4], and [17] all provide a more precise reason for the lack of collapse. For example, in [17], we see that a theory has NIP if and only if all ordered graph indiscernibles collapse to ordered indiscernibles.…”

Section: Collapse Of Indiscerniblesmentioning

confidence: 93%

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On positive local combinatorial dividing-lines in model theory

Guingona

Hill

2018

Arch. Math. Logic

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We introduce the notion of positive local combinatorial dividing-lines in model theory. We show these are equivalently characterized by indecomposable algebraically trivial Fraïssé classes and by complete prime filter classes. We exhibit the relationship between this and collapse-ofindiscernibles dividing-lines. We examine several test cases, including those arising from various classes of hypergraphs.

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Section: Collapse Of Indiscerniblessupporting

confidence: 63%

Section: Collapse Of Indiscerniblesmentioning

confidence: 81%

Section: Collapse Of Indiscerniblesmentioning

confidence: 93%

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On positive local combinatorial dividing-lines in model theory

Guingona

Hill

2018

Arch. Math. Logic

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“…This shows in particular that T +,P r is unstable. We will recall the definition in the proof of Theorem 3.7, but the interested reader may find more in [Sim15] (about NIP) and [CPT14] (on n-dependence).…”

Section: Decidability and Classificationmentioning

confidence: 99%

Decidability and Classification of the Theory of Integers With Primes

Kaplan¹,

Shelah²

2017

J. symb. log.

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We show that under Dickson's conjecture about the distribution of primes in the natural numbers, the theory T h (Z, +, 1, 0, P r) where P r is a predicate for the prime numbers and their negations is decidable, unstable and supersimple. This is in contrast with T h (Z, +, 0, P r, <) which is known to be undecidable by the works of Jockusch, Bateman and Woods.2010 Mathematics Subject Classification. 03C45, 03F30, 03B25, 11A41.

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“…[42] for an introduction to the area). Basic properties of -dependent theories are investigated in [14]. Roughly speaking, -dependence of a theory guarantees that the edge relation of an infinite generic ( + 1)-hypergraph is not definable in its models (see Definition 2.1).…”

Section: Introductionmentioning

confidence: 99%

Onn-dependent groups and fields II

Chernikov

Hempel

2021

Forum of Mathematics, Sigma

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We continue the study of n-dependent groups, fields and related structures, largely motivated by the conjecture that every n-dependent field is dependent. We provide evidence toward this conjecture by showing that every infinite n-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in n-dependent groups, generalizing Shelah’s absoluteness of $G^{00}$ in dependent theories and relative absoluteness of $G^{00}$ in $2$ -dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $2$ -dependent fields with additional structure, showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are $2$ -dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that n-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $2$ -dependence and use it to deduce $2$ -dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory T by a generic predicate is dependent if and only if it is n-dependent for some n, if and only if the algebraic closure in T is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.

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On n-Dependence

Cited by 20 publications

References 21 publications

On positive local combinatorial dividing-lines in model theory

On positive local combinatorial dividing-lines in model theory

Decidability and Classification of the Theory of Integers With Primes

Onn-dependent groups and fields II

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