Compression Schemes, Stable Definable Families, and o-Minimal Structures

Johnson, Hunter R.; Laskowski, M.

doi:10.1007/s00454-009-9201-3

Cited by 14 publications

(16 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This bound is the same as that obtained by Karpinski-Macintyre [49] for o-minimal expansions of the reals, or by Wilkie and by Johnson-Laskowski [47] for all o-minimal structures. The motivating example of a theory which is weakly o-minimal but not ominimal is the theory of real closed valued fields, that is, real closed fields equipped with a predicate for a proper convex valuation ring.…”

Section: Nip Theoriessupporting

confidence: 81%

Vapnik-Chervonenkis density in some theories without the independence property, I

Aschenbrenner

Dolich

Haskell

et al. 2015

Trans. Amer. Math. Soc.

107

View full text Add to dashboard Cite

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and P -minimal theories.Our approach to Theorem 1.1, via definable types, was partly inspired by the use of Puiseux series in [11,81].Let ACVF denote the theory of (non-trivially) valued algebraically closed fields, in the ring language expanded by a predicate for the valuation divisibility. This has completions ACVF (0,0) (for residue characteristic 0), ACVF (0,p) (field characteristic 0, residue characteristic p), and ACVF (p,p) (field characteristic p). Because ACVF (0,0) is interpretable in RCVF, our methods give (non-optimal) density bounds for ACVF (0,0) (Corollary 6.3). However, they give no information on density in the theories ACVF (0,p) and ACVF (p,p) . The problems arise essentially because a definable set in 1-space in ACVF is a finite union of 'Swiss cheeses' but we have no way of choosing a particular Swiss cheese. This means that the definable types technique in our main tool (Theorem 5.7) breaks down. On the other hand, our methods do yield:Theorem 1.2. Suppose M = Q p is the field of p-adic numbers, construed as a firstorder structure in Macintyre's language L p . Then the VC density of every L p -formula ϕ(x; y) is at most 2|y| − 1.

show abstract

Section: Nip Theoriessupporting

confidence: 81%

Vapnik-Chervonenkis density in some theories without the independence property, I

Aschenbrenner

Dolich

Haskell

et al. 2015

Trans. Amer. Math. Soc.

107

View full text Add to dashboard Cite

show abstract

“…It was proved for weakly o-minimal theories in [13] and for dp-minimal theories in [11]. It was proved for weakly o-minimal theories in [13] and for dp-minimal theories in [11].…”

Section: Fact 12mentioning

confidence: 94%

Externally definable sets and dependent pairs II

Chernikov

Simon

2015

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

“…This result can be viewed as a model-theoretic version of the Warmuth conjecture on the existence of compression schemes for VC-families, which was later established in [37]. Special cases of Fact 2.5 were proved earlier for some subclasses of NIP theories including stable [42], o-minimal [28], and dp-minimal [25] theories. Note that this implies Fact 2.3 since, under UDTFS, for every finite set of formulas ∆, every ∆-type over a finite set B is determined by fixing a definition for each ϕ ∈ ∆ with parameters from B, of which there are only polynomially many choices.…”

Section: Preliminaries On Nipmentioning

confidence: 72%

Ramsey Growth in Some Nip Structures

Chernikov¹,

Starchenko²,

Thomas³

2019

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

We investigate bounds in Ramsey's theorem for relations definable in NIP structures. Applying model-theoretic methods to finitary combinatorics, we generalize a theorem of Bukh and Matousek [6] from the semialgebraic case to arbitrary polynomially bounded o-minimal expansions of R, and show that it doesn't hold in Rexp. This provides a new combinatorial characterization of polynomial boundedness for o-minimal structures. We also prove an analog for relations definable in P -minimal structures, in particular for the field of the p-adics. Generalizing [13], we show that in distal structures the upper bound for k-ary definable relations is given by the exponential tower of height k − 1.Recently, this question was investigated in the context of semialgebraic relations, where stronger bounds were obtained. Recall that a set A ⊆ R d is semialgebraic if it is given by a finite Boolean combination of sets of the form {x ∈ R d : f (x) ≥ 0}, where f (x) is a polynomial in d variables with coefficients in R. We say that a semialgebraic set A has description complexity at most t if d ≤ t and A can 2010 Mathematics Subject Classification. Primary 03C45, 05C35, 05D10, 05C25.

show abstract

Compression Schemes, Stable Definable Families, and o-Minimal Structures

Cited by 14 publications

References 16 publications

Vapnik-Chervonenkis density in some theories without the independence property, I

Vapnik-Chervonenkis density in some theories without the independence property, I

Externally definable sets and dependent pairs II

Ramsey Growth in Some Nip Structures

Contact Info

Product

Resources

About