Externally definable sets and dependent pairs

Chernikov, Artem; Simon, Pierre

doi:10.1007/s11856-012-0061-9

Cited by 57 publications

(79 citation statements)

References 17 publications

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“…Thus the L-formula ψ( x, y) has the IP, a contradiction. By Proposition 3.2 every formula in (M, H) is equivalent to a boolean combination of existential formulas over H. This fact together with Theorem 2.4 [12] shows that T h(M, H) also has NIP.…”

Section: 3mentioning

confidence: 72%

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Geometric structures with a dense independent subset

Berenstein

Vassiliev

2015

Sel. Math. New Ser.

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Abstract. We generalize the work of [13] on expansions of o-minimal structures with dense independent subsets, to the setting of geometric structures. We introduce the notion of an H-structure of a geometric theory T , show that H-structures exist and are elementarily equivalent, and establish some basic properties of the resulting complete theory T ind , including quantifier elimination down to "H-bounded" formulas, and a description of definable sets and algebraic closure. We show that if T is strongly minimal, supersimple of SUrank 1, or superrosy of thorn rank 1, then T ind is ω-stable, supersimple, and superrosy, respectively, and its U-/SU-/thorn rank is either 1 (if T is trivial) or ω (if T is non-trivial). In the supersimple SU-rank 1 case, we obtain a description of forking and canonical bases in T ind . We also show that if T is (strongly) dependent, then so is T ind , and if T is non-trivial of finite dp-rank, then T ind has dp-rank greater than n for every n < ω, but bounded by ω. In the stable case, we also partially solve the question of whether any group definable in T ind comes from a group definable in T .

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Section: 3mentioning

confidence: 72%

“…Proof. We apply the criterion from [12,Thm 2.4]. We begin by showing that every formula φ( x, y) has NIP over H( x).…”

Section: 3mentioning

confidence: 99%

Geometric structures with a dense independent subset

Berenstein

Vassiliev

2015

Sel. Math. New Ser.

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“…On the one hand, we complement the result in [8] by showing that naming a small indiscernible sequence of arbitrary order type is bounded and preserves NIP. On the one hand, we complement the result in [8] by showing that naming a small indiscernible sequence of arbitrary order type is bounded and preserves NIP.…”

Section: Introductionmentioning

confidence: 72%

“…In [8] we gave a general condition for the expansion to be NIP: it is enough that the theory of the pair is bounded, i.e., eliminates quantifiers down to the predicate, and the induced structure on the predicate is NIP. In [8] we gave a general condition for the expansion to be NIP: it is enough that the theory of the pair is bounded, i.e., eliminates quantifiers down to the predicate, and the induced structure on the predicate is NIP.…”

Section: Introductionmentioning

confidence: 99%

“…If it is not, then there is some formula giving the order on the sequence, and by Fact 1, φ-types over I are definable using this order (see [8,Section 3.1]). In particular, if I is a complete linear order and φ(x 0 , ..., x n ; y) is NIP, then all φ-types over I are definable, possibly after adding finitely many elements extending I on both sides.…”

Section: Introductionmentioning

confidence: 99%

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Externally definable sets and dependent pairs II

Chernikov

Simon

2015

Trans. Amer. Math. Soc.

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We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in a distal theory; naming an arbitrary small indiscernible sequence preserves NIP, while naming a large one doesn't; there are models of NIP theories over which all 1-types are definable, but not all n-types.

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Generic trivializations of geometric theories

Berenstein

Vassiliev

2014

Mathematical Logic Qtrly

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Abstract. We study the theory T * of the structure induced by parameter free formulas on a dense algebraically independent subset of a model of a geometric theory T . We show that while being a trivial geometric theory, T * inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, N IP and N T P 2 . In particular, we show that T is strongly minimal, supersimple of SU-rank 1, or NIP exactly when so is T * . We show that if T is superrosy of thorn rank 1, then so is T * , and that the converse holds if T satis es acl = dcl.

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Externally definable sets and dependent pairs

Cited by 57 publications

References 17 publications

Geometric structures with a dense independent subset

Geometric structures with a dense independent subset

Externally definable sets and dependent pairs II

Generic trivializations of geometric theories

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