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## Vapnik-Chervonenkis density in some theories without the independence property (2011)

Citations: | 10 - 0 self |

### Citations

380 |
den Dries, Tame topology and o-minimal structures (Cambridge
- van
- 1998
(Show Context)
Citation Context ...h partitioned L-formula ϕ(x; y) with |x| = 1 defines a finite union of convex sets. (See [61] for more on this notion, which generalizes the probably more familiar concept of an o-minimal theory, cf. =-=[25]-=-.) Theorem 1.1. Suppose L contains a binary relation symbol “<”, interpreted in M as a linear ordering. If T = Th(M) is weakly o-minimal, then every L-formula ϕ(x; y) has VC density at most n = |y| in... |

316 |
On the density of families of sets
- Sauer
- 1972
(Show Context)
Citation Context ...ction of S. Clearly 0 ≤ πS(n) ≤ 2n for every n, and if S is not a VC class, then πS(n) = 2n for every n. However, if S is a VC class, of VC dimension d say, then by a fundamental observation of Sauer =-=[83]-=- (independently made in [87] and, implicitly, in [98]), the function n 7→ πS(n) is bounded above by a polynomial in n of degree d. (In fact, for d, n ≥ 1 one has πS(n) ≤ (en/d)d, where e is the base o... |

204 |
On a problem of K.
- Kovari, Sos, et al.
- 1954
(Show Context)
Citation Context ...and s we denote by Kr,s := ( [r], [s], [r] × [s]) the complete bigraph with the vertex set [r] ∪ [s]. The following is a fundamental fact about finite bigraphs: Theorem 4.2 (Kővári, Sós and Turán =-=[51]-=-). Let r ≤ s be positive integers. There exists a real number C = C(r, s) such that every finite bigraph G which does not contain Kr,s as a sub-bigraph has at most C |V (G)|2−1/r edges. (In fact, a mo... |

191 | Almost optimal set covers in finite VC-dimension
- Brönnimann, Goodrich
- 1995
(Show Context)
Citation Context ... sup n→∞ log πS(n) logn . We also define VC(∅) := vc(∅) := −1. Then vc(S) ≤ VC(S) by Lemma 2.1, and vc(S) <∞ iff VC(S) <∞. The VC density of S is also known as the real density [7] or the VC exponent =-=[17]-=- of S. It is related to the combinatorial dimension of S introduced by Blei [8] and to compression schemes for S [47]. Example. Suppose S = (X≤d). Then the inequality in the statement of Lemma 2.1 is ... |

174 |
Model theory, Encyclopedia of Mathematics and its
- Hodges
- 1993
(Show Context)
Citation Context ...ions and basic facts. We hope that the present paper (unlike its sequel [6]) can be read with only little technical knowledge of model theory beyond basic first order logic. The first few chapters of =-=[42]-=- or [63] or similar texts should provide sufficient background for a prospective reader. 1.1. VC dimension and VC density. Let X be an infinite set and S be a non-empty collection of subsets of X . Gi... |

112 |
den Dries, p-adic and real subanalytic sets
- Denef, van
- 1988
(Show Context)
Citation Context ...cture in Macintyre’s language Lp. Then the VC density of every Lp-formula ϕ(x; y) is at most 2|y| − 1. The same result holds for the subanalytic expansions of Qp considered by Denef and van den Dries =-=[22]-=-. (Theorem 7.2 and Remark 7.9.) Key tools available here, but not in the case of ACVF, are cell decomposition and the existence of definable Skolem functions. We do not know whether the bound in Theor... |

111 | Sphere packing numbers for subsets of the Booleann-cube with bounded Vapnik-Chervonenkis dimension
- Haussler
- 1991
(Show Context)
Citation Context ... the VC dimension) is the decisive measure for the combinatorial complexity of a family of sets. For example, the VC density of S governs the size of packings in S with respect to the Hamming metric (=-=[41]-=-, see also [64, Lemma 2.1]), and is intimately related to the notions of entropic dimension [7] and discrepancy [68]. We refer to the surveys [65, 33] for uses of VC density in combinatorics. 1.2. VC ... |

97 | Model Theory and Modules, - Prest - 1988 |

90 | A course on empirical processes - Dudley - 1984 |

85 | Lattice Theory, 3rd ed., - Birkhoff - 1967 |

75 | A partition calculus in set theory,
- Erdos, Rado
- 1956
(Show Context)
Citation Context ... not contain Kr,s as a sub-bigraph, for some r, s ≥ 1, then the bigraph G¬ϕ associated to ¬ϕ does contain Kt,t, for every t ≥ 1: by an analogue of Ramsey’s Theorem for bigraphs due to Erdős and Rado =-=[32]-=-, for every t there exists an n such that for all bigraphs G with |V (G)| ≥ n, one of G, ¬G contains Kt,t as a sub-bigraph. Hence in this case the VC density of the formula ¬̂ϕ associated to ¬ϕ equals... |

75 |
The Strange Logic of Random Graphs
- Spencer
- 2001
(Show Context)
Citation Context ...of the question posed above.) However, it turns out that vc(ϕ) is an integer: Lemma 4.9. vc(ϕ) = ⌊1/α⌋. Before we give the proof, we recall some basic facts about the theory Tα; our main reference is =-=[94]-=-. We let G be a model of Tα. A rooted graph is a pair (R,H) where H is a finite graph and R a proper subset of its set of vertices; the elements of R will be called roots. We consider each finite non-... |

71 | Linear equations in variables which lie in a multiplicative group
- Evertse, Schlickewei, et al.
(Show Context)
Citation Context ... ∑ i∈I aigi 6= 0 for each non-empty I ⊆ [n]. (Examples for multiplicative groups with the Mann property include all finite-rank subgroups of K× if K is algebraically closed of characteristic zero, by =-=[12]-=-.) In [10], van den Dries and Günaydın study the model theory of pairs (K,G) (in the language of fields expanded by a unary predicate symbol) where K is algebraically closed or real closed and G has ... |

60 |
Zero-one laws for sparse random graphs,
- Shelah, Spencer
- 1988
(Show Context)
Citation Context ... Lgrstructures are nothing but the (directed) graphs (with E interpreted as the edge relation). Given a graph G we denote by V (G) its set of vertices and by E(G) its set of edges. Spencer and Shelah =-=[91]-=- established a 0-1-law for Lgr-sentences about random (symmetric, loopless) graphs with n vertices and edge probability n−α, where α is an irrational number between 0 and 1. We denote the resulting co... |

54 | Polynomial bounds for VC dimension of sigmoidal and general pfaffian neural networks
- Karpinski, Macintyre
- 1995
(Show Context)
Citation Context ... several examples of classes of sets with finite VC dimension, by noting well-known examples of theories without the independence property. This line of thought was pursued by Karpinski and Macintyre =-=[49]-=-, who calculated explicit bounds on the VC dimension of definable families of sets in some o-minimal structures (with an eye towards applications to neural networks), which were polynomial in the numb... |

48 |
Cours de théorie des modèles (Nur Al-Mantiq Wal-Ma’rifah
- Poizat
- 1985
(Show Context)
Citation Context ...rty if some L-formula has the independence property for M , and not to have the independence property (or to be NIP or dependent) otherwise. By a classical result of Shelah [86] (with other proofs in =-=[52, 55, 80]-=-), for M to be NIP it is actually sufficient that no formula ϕ(x; y) with |x| = 1 has the independence property for M . NIP is implied by (but not equivalent to) another prominent tameness condition o... |

40 | Vapnik-chervonenkis classes of definable sets
- Laskowski
- 1992
(Show Context)
Citation Context ...ferences 56 1. Introduction The notion of VC dimension, which arose in probability theory in the work of Vapnik and Chervonenkis [98], was first drawn to the attention of model-theorists by Laskowski =-=[55]-=-, who observed that a complete first-order theory does not have the independence property (as introduced by Shelah [86]) if and only if, in each model, each definable family of sets has finite VC dime... |

40 | Weakly o-minimal structures and real closed fields
- Macpherson, Marker, et al.
(Show Context)
Citation Context ...ally, separably closed) fields, differentially closed fields, modules, or free groups furnish examples of NIP structures. Furthermore, o-minimal (or more generally, weakly o-minimal) theories are NIP =-=[55, 61]-=-. By [36] any ordered abelian group has NIP theory. Certain important theories of henselian valued fields are NIP, for example, the completions of the theory of algebraically closed valued fields and ... |

39 |
Unit distances in the Euclidean plane
- Spencer, Szemerédi, et al.
- 1984
(Show Context)
Citation Context ..., so SRϕ is the collection of circles with radius 1 in the plane, and E(GRϕ ) is the set of incidences between points in R2 and circles of radius 1. Then an analogue of the Szémeredi-Trotter Theorem =-=[95]-=- (or a more general result due to Pach and Sharir [73] on families of simple plane curves) and (4.1) yields vcR(ϕ̂) ≤ 43 . (However, it is unknown whether this bound is sharp, cf. [74, Section 2].) 4.... |

33 |
Repeated angles in the plane and related problems,
- Pach, Sharir
- 1992
(Show Context)
Citation Context ...= (R, 0, 1,+,−,×, <). For this we use another one of the rare examples (besides the Szémeredi-Trotter Theorem) where tight bounds on the number of incidences are known: Theorem 4.13 (Pach and Sharir =-=[72]-=-). Let α be a real number with 0 < α < π. The maximum number of times that α occurs as an angle among the ordered triples of t points in the plane is O(t2 log t). Furthermore, suppose tan(α) ∈ Q√d whe... |

32 |
Rigid subanalytic sets,
- Lipshitz
- 1993
(Show Context)
Citation Context ...er elimination in ACVF; see [43]). Conversely, every valued field with C-minimal elementary theory is algebraically closed [38]. Moreover, the rigid analytic expansions of ACVF introduced by Lipshitz =-=[57]-=- are C-minimal [58]. Example 3.11. Let R be a ring and suppose L = LR is the language of R-modules. (In this paper, “R-module” always means “left R-module.”) Suppose M is an R-module, construed as an ... |

31 | On the number of cells defined by a family of polynomials on a variety
- Basu, Pollack, et al.
- 1996
(Show Context)
Citation Context ... these methods do not readily adapt may be found in [5, 54]. VC DENSITY IN SOME NIP THEORIES, I 5 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 res... |

27 |
Sums versus products in number theory, algebra and Erdos geometry.
- Elekes
- 2002
(Show Context)
Citation Context ...cular VC(ϕ) = vc(ϕ) = 2. A lower bound on vc(ϕ̂) is given by: Lemma 4.5. Suppose K has characteristic 0. Then vc(ϕ̂) ≥ 43 . Proof. This is due to Erdős, with the following simpler argument by Elekes =-=[31]-=-: let k be a positive integer, t = 4k3, and consider the subsets P := { (η, ξ) : η = 0, 1, . . . , k − 1, ξ = 0, 1, . . . , 4k2 − 1} L := { (a, b) : a = 0, 1, . . . , 2k − 1, b = 0, 1, . . . , 2k2 − 1... |

26 |
Cell decompositions of C-minimal structures
- Haskell, Macpherson
- 1994
(Show Context)
Citation Context ...efinition of C-minimality used in the previous example agrees (for expansions of valued fields) with the one in [44]; this definition is slightly more restrictive than the original one, introduced in =-=[38, 62]-=-. Every completion of the Ldiv-theory ACVF of non-trivially valued algebraically closed fields is C-minimal (essentially by A. Robinson’s quantifier elimination in ACVF; see [43]). Conversely, every v... |

25 |
Discrepancy and approximations for bounded VC-dimension.
- Matousek, Welzl, et al.
- 1993
(Show Context)
Citation Context ... density of S governs the size of packings in S with respect to the Hamming metric ([41], see also [64, Lemma 2.1]), and is intimately related to the notions of entropic dimension [7] and discrepancy =-=[68]-=-. We refer to the surveys [65, 33] for uses of VC density in combinatorics. 1.2. VC dimension and VC density of formulas. Let L be a first-order language. In an L-structure M , a natural way to genera... |

23 |
the f.c.p., and superstability; model theoretic properties of formulas in first order theory
- Stability
- 1971
(Show Context)
Citation Context ...rvonenkis [98], was first drawn to the attention of model-theorists by Laskowski [55], who observed that a complete first-order theory does not have the independence property (as introduced by Shelah =-=[86]-=-) if and only if, in each model, each definable family of sets has finite VC dimension. With this observation, Laskowski easily gave several examples of classes of sets with finite VC dimension, by no... |

23 | 0-categorical, ℵ0-stable structures - Cherlin, Harrington, et al. - 1985 |

22 | Integration in valued fields,
- Hrushovski, Kazhdan
- 2006
(Show Context)
Citation Context ...s a Boolean combination of at most N balls in K. Thus vcT (1) = 1 by Example 2.12. The definition of C-minimality used in the previous example agrees (for expansions of valued fields) with the one in =-=[44]-=-; this definition is slightly more restrictive than the original one, introduced in [38, 62]. Every completion of the Ldiv-theory ACVF of non-trivially valued algebraically closed fields is C-minimal ... |

22 | The Szemeredi-Trotter theorem in the complex plane. arXiv:math/0305283v4
- Toth
- 2003
(Show Context)
Citation Context ...zation of a famous theorem of Szémeredi and Trotter [96] (although a weaker version of this theorem from [93], with a somewhat simpler proof, would also suffice for our purposes): Theorem 4.7 (Tóth =-=[97]-=-). There exists a real number C such that for all m,n > 0 there are at most C(m2/3n2/3 +m+ n) incidences among m points and n lines in the affine plane over C. Proof of Proposition 4.6. The lower boun... |

20 | Analytic p-adic cell decomposition and integrals - Cluckers |

20 |
Weakly normal groups
- Hrushovski, Pillay
- 1987
(Show Context)
Citation Context ...nable subset of G is a Boolean combination of cosets of acleq(∅)-definable subgroups of G. (This condition holds, in particular, if T satisfies the model-theoretic condition known as 1-basedness, cf. =-=[45]-=-.) By Example 2.14, if the collection of acleq(∅)- definable subgroups of G has breadth at most d (in particular, by Example 2.16, if it has height at most d), then we have vcT (1) ≤ d. Here is a part... |

20 | Tight upper bounds for the discrepancy of half-spaces. - Matousek - 1995 |

20 |
On the number of cells defined by a set of polynomials,
- Pollack, Roy
- 1993
(Show Context)
Citation Context ... these methods do not readily adapt may be found in [5, 54]. VC DENSITY IN SOME NIP THEORIES, I 5 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 res... |

19 | One-dimensional p-adic subanalytic sets
- Dries, Haskell, et al.
- 1999
(Show Context)
Citation Context ...ifier-free definable just using the language Lp. (In fact, the setting of [39] also allowed for p-adically closed fields of arbitrary fixed p-rank, and our methods here could be adjusted to that.) By =-=[26]-=-, a motivating example of a P -minimal theory is the theory pCFan first investigated by Denef and van den Dries [22]. This is the theory of the p-adic numbers equipped with, for every n > 0 and power ... |

19 |
CERVONENKIS:' The Uniform convergence of frequencies of the apperance of events to their probabilities
- VAPNIK, J
- 1971
(Show Context)
Citation Context ... and Variants 42 7. A Strengthening of VC d, and P -adic Examples 47 References 56 1. Introduction The notion of VC dimension, which arose in probability theory in the work of Vapnik and Chervonenkis =-=[98]-=-, was first drawn to the attention of model-theorists by Laskowski [55], who observed that a complete first-order theory does not have the independence property (as introduced by Shelah [86]) if and o... |

18 |
Canonical forms for definable subsets of algebraically closed and real closed valued fields
- Holly
- 1995
(Show Context)
Citation Context ...ne, introduced in [38, 62]. Every completion of the Ldiv-theory ACVF of non-trivially valued algebraically closed fields is C-minimal (essentially by A. Robinson’s quantifier elimination in ACVF; see =-=[43]-=-). Conversely, every valued field with C-minimal elementary theory is algebraically closed [38]. Moreover, the rigid analytic expansions of ACVF introduced by Lipshitz [57] are C-minimal [58]. Example... |

17 |
Model theory, Graduate Texts
- Marker
- 2002
(Show Context)
Citation Context ... basic facts. We hope that the present paper (unlike its sequel [6]) can be read with only little technical knowledge of model theory beyond basic first order logic. The first few chapters of [42] or =-=[63]-=- or similar texts should provide sufficient background for a prospective reader. 1.1. VC dimension and VC density. Let X be an infinite set and S be a non-empty collection of subsets of X . Given A ⊆ ... |

16 |
corps faiblement algébriquement clos non séparablement clos ont la propriété d’indépendance’, in Model theory of algebra and arithmetic
- Duret, ‘Les
- 1980
(Show Context)
Citation Context ... Kaplansky fields of characteristic (p, p) are NIP iff their residue fields are NIP [13, 12]. On the other hand, each pseudofinite field (infinite model of the theory of all finite fields) is not NIP =-=[29]-=-, since it defines the (Rado) random graph. 1.4. Uniform bounds on VC density. This paper is motivated by the following question: Given a NIP theory T , can one find an upper bound, in terms of n only... |

16 |
Presburger arithmetic and recognizability of sets of natural numbers by automata: New proofs of Cobham’s theorem and Semenov’s theorem, Annals of Pure and Applied Logic 77
- Michaux, Villemaire
- 1996
(Show Context)
Citation Context ...hen Sϕ is eventually finite. For the benefit of the reader we now indicate the details of the proof of Proposition 6.6. Let Z be a proper expansion of (Z, <,+). By a theorem of Michaux and Villemaire =-=[69]-=- (and an easy extra argument, given in the proof of [14, Theorem 10]), there is a subset U of Z which is definable in Z but not definable in (Z, <,+). By Simon’s lemma, the family {a + U : a ∈ Z} is e... |

16 |
On ω1-categorical theories of abelian groups
- Macintyre
- 1971
(Show Context)
Citation Context ... subsection with treating another simple special case: Corollary 5.26. Suppose T is ℵ1-categorical. Then vcT (m) = m for each m. (In particular, T is dp-minimal.) Proof By a theorem of Macintyre (see =-=[26]-=- or [17, Theorem A.2.12]), A = B ⊕C where B is finite and either C is divisible and C[p] is finite for each prime p; or C = Z(pn)(α) for some prime p, some n > 0, and some (infinite) cardinal α. In th... |

15 | Traces of finite sets: extremal problems and geometric applications,” Extremal Problems for Finite Sets (Visegrad,
- Furedi, Pach
- 1991
(Show Context)
Citation Context ... of packings in S with respect to the Hamming metric ([41], see also [64, Lemma 2.1]), and is intimately related to the notions of entropic dimension [7] and discrepancy [68]. We refer to the surveys =-=[65, 33]-=- for uses of VC density in combinatorics. 1.2. VC dimension and VC density of formulas. Let L be a first-order language. In an L-structure M , a natural way to generate a collection of subsets of Mm i... |

14 | The Vapnik-Chervonenkis dimension of a random graph
- Anthony, Brightwell, et al.
- 1995
(Show Context)
Citation Context ...raph. We remark that a similar analysis shows that the simpler formula E(x, y) also has VC density ⌊1/α⌋ in Tα. We chose ϕ as above because it allows us to compare Lemma 4.9 with the main result of =-=[4]-=-, where the precise value of the VC dimension of ϕ (as it depends on α) is computed. In particular, [4, Corollary 8] shows that if 0 < α < 93650 then ⌊1/α⌋+ 3 ≤ VC(ϕ) ≤ ⌊1/α+ 3(α+ 1)⌋. 4.4. Shatter fu... |

14 |
den Dries, Closed asymptotic couples
- Aschenbrenner, van
(Show Context)
Citation Context ...ensity bounds for this and certain other weakly o-minimal expansions of real closed fields [40]. Some interesting weakly o-minimal theories to which these methods do not readily adapt may be found in =-=[5, 54]-=-. VC DENSITY IN SOME NIP THEORIES, I 5 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) val... |

14 |
Elimination of quantifiers for ordered valuation rings
- Dickmann
- 1987
(Show Context)
Citation Context ...a binary relation symbol. Let RCVF denote the theory of real closed fields equipped with a proper convex valuation ring, parsed in the language Ldiv,<. The following corollary is now immediate, as by =-=[23]-=-, RCVF is weakly o-minimal. Corollary 6.2. Let K |= RCVF. Then any finite set ∆(x; y) of Ldiv,<-formulas has dual VC density at most |x| in K. This result in turn yields a VC density bound for algebra... |

14 | One-dimensional fibers of rigid subanalytic sets
- Lipshitz, Robinson
- 1998
(Show Context)
Citation Context ...CVF; see [43]). Conversely, every valued field with C-minimal elementary theory is algebraically closed [38]. Moreover, the rigid analytic expansions of ACVF introduced by Lipshitz [57] are C-minimal =-=[58]-=-. Example 3.11. Let R be a ring and suppose L = LR is the language of R-modules. (In this paper, “R-module” always means “left R-module.”) Suppose M is an R-module, construed as an LR-structure in the... |

14 |
A combinatorial distinction between the Euclidean and projective planes
- Szemerédi, Trotter
- 1983
(Show Context)
Citation Context ... 0. Then vc(ϕ̂) = 43 . (2) Suppose K has positive characteristic. Then vc(ϕ̂) = 32 . In the proof of this proposition we use the following generalization of a famous theorem of Szémeredi and Trotter =-=[96]-=- (although a weaker version of this theorem from [93], with a somewhat simpler proof, would also suffice for our purposes): Theorem 4.7 (Tóth [97]). There exists a real number C such that for all m,n... |

13 |
Theories controlled by formulas of Vapnik-Chervonenkis codimension 1
- Adler
- 2008
(Show Context)
Citation Context ...hus if S is the family of all Boolean combinations of at most N balls in K, for some N ∈ N, then π∗S(t) = O(t). The preceding examples can be subsumed under the following general example (inspired by =-=[2]-=-): Example 2.13. A family B of subsets of X is said to be directed if B has breadth 1; i.e., for all B,B′ ∈ B with B ∩B′ 6= ∅ one has B ⊆ B′ or B′ ⊆ B. If B ⊆ 2X is directed and S is the family of Boo... |

13 |
first order theories, continued
- Dependent
(Show Context)
Citation Context ...) and every c ∈ (M∗)|z|. The Shelah expansion of M is the expansion of M to an LSh-structure MSh where each predicate symbol Rψ,c(x) as before is interpreted by M |x| ∩ ψM∗((M∗)|x|; c). Shelah showed =-=[89]-=- (with another proof given in [20]) that if T is NIP then T Sh = Th(MSh) admits 24 ASCHENBRENNER, DOLICH, HASKELL, MACPHERSON, AND STARCHENKO quantifier elimination and is also NIP. This provides an i... |

13 | On dp-minimal ordered structures
- Simon
- 2009
(Show Context)
Citation Context ... expansions. One can strengthen this statement: Proposition 6.6. No proper expansion of (Z, <,+) is dp-minimal. The proof is the same as in [14], replacing the use of [14, Theorem 7] by a result from =-=[92]-=-; we state the latter employing some convenient terminology from [14]: Let (X,<) be a linearly ordered set. We say that two subsets S, T of X are eventually equal (in symbols: S ≈ T ) if there is some... |

13 |
The fields of real and complex numbers with a small multiplicative group
- Dries, Günaydın
(Show Context)
Citation Context ...i 6= 0 for each non-empty I ⊆ [n]. (Examples for multiplicative groups with the Mann property include all finite-rank subgroups of K× if K is algebraically closed of characteristic zero, by [12].) In =-=[10]-=-, van den Dries and Günaydın study the model theory of pairs (K,G) (in the language of fields expanded by a unary predicate symbol) where K is algebraically closed or real closed and G has the Mann p... |

12 |
On variants of o-minimality, Ann
- Macpherson, Steinhorn
- 1996
(Show Context)
Citation Context ...efinition of C-minimality used in the previous example agrees (for expansions of valued fields) with the one in [44]; this definition is slightly more restrictive than the original one, introduced in =-=[38, 62]-=-. Every completion of the Ldiv-theory ACVF of non-trivially valued algebraically closed fields is C-minimal (essentially by A. Robinson’s quantifier elimination in ACVF; see [43]). Conversely, every v... |

12 |
On dp-minimality, strong dependence, and weight
- Onshuus, Usvyatsov
(Show Context)
Citation Context ... be found in [2, 24]. See also [48] for a generalization of Corollary 5.13 to a bound on “dp-rank” in terms of VC density. A characterization of dp-minimal theories among stable theories was given in =-=[71]-=-. The main result of [34] is that every dp-minimal theory T has UDTFS. In particular, by Corollary 5.13, every vc-minimal T has UDTFS. (Actually, [34, Theorem 3.14] gives a more precise result: if ϕ(x... |

11 |
Unavoidable traces of set systems
- BALOGH, BOLLOBÁS
(Show Context)
Citation Context ...X, Y be infinite sets and Φ ⊆ X×Y be a relation. If vc(Φ) > 0 then Φ is unstable, or at least one of Φ or ¬Φ has infinite breadth. At the root of Proposition 2.20 is a theorem of Balogh and Bollobás =-=[10]-=-, which we explain first. For this we need some additional terminology: Let (X,S) and (X ′,S ′) be set systems. We say that (X,S) contains (X ′,S ′) as a trace if there exists an injective map f : X ′... |

11 | Regularity partitions and the topology of graphons.
- Lovasz, Szegedy
- 2010
(Show Context)
Citation Context ...hatter function is not asymptotic to a real power function. Throughout this section L is a first-order language and M is an L-structure. 4.1. Associating a bigraph to a partitioned formula. We follow =-=[59]-=- and make a distinction between bipartite graphs and bigraphs. A bipartite graph is a graph (V,E) whose set V of vertices can be partitioned into two classes such that all edges connect vertices in di... |

10 |
Types dans les corps valués munis d’applications coefficients
- Bélair
- 1999
(Show Context)
Citation Context ... respectively). In fact, in the language of rings with a predicate for the valuation ring, an unramified henselian valued field of characteristic (0, p) is NIP if and only if its residue field is NIP =-=[12]-=-. Similarly, henselian valued fields 4 ASCHENBRENNER, DOLICH, HASKELL, MACPHERSON, AND STARCHENKO of characteristic (0, 0) and algebraically maximal Kaplansky fields of characteristic (p, p) are NIP i... |

10 |
Quasi-o-minimal structures
- Belegradek, Peterzil, et al.
(Show Context)
Citation Context ...n to quasi-o-minimal theories: T is said to be quasi-o-minimal if for any M |= T , any definable subset of M is a finite Boolean combination of singletons, intervals in M , and ∅-definable sets. (See =-=[14]-=-.) Theorem 6.4. Assume that T is quasi-o-minimal. Then T has the VC1 property, and hence vcT (n) = n for each n. 44 ASCHENBRENNER, DOLICH, HASKELL, MACPHERSON, AND STARCHENKO Proof. Let M |= T . Fix a... |

10 | Stable theories with a new predicate - Casanovas, Ziegler |

10 |
The theory of ordered abelian groups does not have the independence property
- Gurevich, Schmitt
- 1984
(Show Context)
Citation Context ...ly closed) fields, differentially closed fields, modules, or free groups furnish examples of NIP structures. Furthermore, o-minimal (or more generally, weakly o-minimal) theories are NIP [55, 61]. By =-=[36]-=- any ordered abelian group has NIP theory. Certain important theories of henselian valued fields are NIP, for example, the completions of the theory of algebraically closed valued fields and the theor... |

9 |
On extensions of Presburger arithmetic
- Cherlin, Point
- 1986
(Show Context)
Citation Context ...(Z, <,+) by the set of factorials or by the set of Fibonacci numbers. In all these examples, the corresponding expansion of (Z, <,+) has quantifier elimination in a natural expansion of {<,+, U} (see =-=[19, 78]-=-) and is NIP (as will be shown elsewhere). 6.3. Weakly quasi-o-minimal theories. In [53], T is called weakly quasi-o-minimal if for any M |= T , any definable subset of M is a finite Boolean combinati... |

9 |
Elimination of quantifiers for certain ordered and lattice-ordered abelian groups, Bulletin de la Société Mathématique de Belgique
- Weispfenning
(Show Context)
Citation Context ...ordering on Zn ×Q. Proof. Each of the examples has quasi-o-minimal theory: For (1) this was noted in [14, Section 1], and for (2) and (3) this is proved (based on a quantifier-elimination result from =-=[99]-=-) in [15, Theorem 15]. The corollary now follows from Theorem 6.4. Remark. The ordered abelian groups in (2) and (3) of the previous corollary are typical for quasi-o-minimal groups. Here and below,... |

8 |
The model-theoretic content of Lang's conjecture
- Pillay
- 1998
(Show Context)
Citation Context ...ounded formula, and let λ ≥ |L| be a cardinal. If both M and Aind are λ-stable then (M , A) is λ-stable. (See [18, Corollary 5.4 and Proposition 3.1]; part (1) had actually first been shown by Pillay =-=[76]-=-.) Consider now, slightly more general than necessary, an arbitrary field K, and let LK = {0,+, (λ· )λ∈K} be the language of K-vector spaces. Let M be an infinitedimensional K-vector space. Then M , c... |

8 |
Generically stable and smooth measures
- Hrushovski, Pillay, et al.
- 2010
(Show Context)
Citation Context ...ncompasses, among other examples, all weakly o-minimal theories and many interesting theories of expansions of valued fields. In recent years, not least due to the efforts of Anand Pillay (see, e.g., =-=[18; 20; 21]-=-), there has been some progress in extending the reach of the highly developed methods of stability theory into this wider realm of NIP theories. In this paper we establish uniform bounds, in terms of... |

7 | On uniform definability of types over finite sets
- Guingona
(Show Context)
Citation Context ... which allows the counting of the number of ∆(x;B)-types over finite parameter sets B. The definition of the VC d property rests on a “uniform” variant of the notion of definable type, originating in =-=[13]-=-: Definition 2.4. We say that ∆ has uniform definability of types over finite sets (abbreviated as UDTFS ) inM with d parameters if there are finitely many families Di = ( dϕ,i(y; y1, . . . , yd) ) ϕ∈... |

7 |
On Goldie and dual Goldie dimensions
- Grzeszczuk, PuczyÃlowski
(Show Context)
Citation Context ... join-independent subset of size n, if there is such an n; otherwise we set Gdim(L) =∞. The Goldie dimension of the dual L∗ of L is called the dual Goldie dimension of L and denoted by Gdim∗(L). (See =-=[15]-=-.) Lemma 4.1. Suppose L is modular. Then breadth(L) ≥ max{Gdim(L),Gdim∗(L)}. VC Density in some NIP Theories, II 19 Proof Since the lattices L and L∗ have the same breadth, it suffices to prove that b... |

7 | On Lascar rank and Morley rank of definable groups in differentially closed fields - Pillay, Pong |

6 | C.: Compression Schemes, stable definable families, and o-minimal structures Discrete Comput Geom (2010
- Johnson, Laskowski
- 2009
(Show Context)
Citation Context ...ndence property). The bound in the o-minimal case in [50] was established independently, using a more combinatorial approach, by Wilkie (unpublished), and more recently, also by Johnson and Laskowski =-=[47]-=-. Date: September 2011. 1 2 ASCHENBRENNER, DOLICH, HASKELL, MACPHERSON, AND STARCHENKO In this paper we give a sufficient criterion (Theorem 5.7) on a first-order theory for the VC density of a defina... |

6 | Abelian groups with contractions, II: Weak o-minimality
- Kuhlmann
- 1995
(Show Context)
Citation Context ...ensity bounds for this and certain other weakly o-minimal expansions of real closed fields [40]. Some interesting weakly o-minimal theories to which these methods do not readily adapt may be found in =-=[5, 54]-=-. VC DENSITY IN SOME NIP THEORIES, I 5 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) val... |

5 |
Cell decomposition for P -minimal fields
- Mourgues
(Show Context)
Citation Context ...)(ci(b)), where i = 1, . . . , βd, are exactly the balls of distance at most d to B. The assumption of definable Skolem functions also guarantees that any model of the theory T has cell decomposition =-=[70]-=-. Let ∆(x; y) be a finite set of L-formulas, where |x| = 1, closed under negation. Then there are integers N,n > 0, and for each i = 1, . . . , N there are ∅-definable functions fi, gi, ci : K |y| → K... |

5 | General Lattice Theory, 2nd ed., Birkhäuser - Grätzer - 1998 |

4 |
Dp-minimal theories: basic facts and examples
- Dolich, Lippel, et al.
- 2011
(Show Context)
Citation Context ...und on a number of types also illuminates the connection with a strengthening of the NIP concept, which is that of dp-minimality. (See Section 5.3 below for a definition.) Dolich, Goodrick and Lippel =-=[24]-=- have observed that, if, in a theory, the dual VC density of any L-formula in a single object variable is less than 2, then the theory in question is dp-minimal. (No counterexample to the converse of ... |

4 |
On decidable extensions of Presburger arithmetic: from A. Bertrand numeration systems to Pisot numbers
- Point
(Show Context)
Citation Context ...(Z, <,+) by the set of factorials or by the set of Fibonacci numbers. In all these examples, the corresponding expansion of (Z, <,+) has quantifier elimination in a natural expansion of {<,+, U} (see =-=[19, 78]-=-) and is NIP (as will be shown elsewhere). 6.3. Weakly quasi-o-minimal theories. In [53], T is called weakly quasi-o-minimal if for any M |= T , any definable subset of M is a finite Boolean combinati... |

4 |
An incidence theorem in higher dimensions, preprint (2011), available online at http://front.math.ucdavis.edu/1103.2926
- Solymosi, Tao
(Show Context)
Citation Context ...racteristic. Then vc(ϕ̂) = 32 . In the proof of this proposition we use the following generalization of a famous theorem of Szémeredi and Trotter [96] (although a weaker version of this theorem from =-=[93]-=-, with a somewhat simpler proof, would also suffice for our purposes): Theorem 4.7 (Tóth [97]). There exists a real number C such that for all m,n > 0 there are at most C(m2/3n2/3 +m+ n) incidences a... |

4 |
αT is finite for ℵ1-categorical T
- Baldwin
- 1973
(Show Context)
Citation Context ...-algebraic 1-types are non-orthogonal). Then T satisfies the hypothesis of the previous proposition with a single Di. (The fact that MR(T ) is finite for ℵ1-categorical countable T was first shown in =-=[3]-=-.) 3.4 Applications. Together with the remarks in the previous subsection, we can now draw some immediate consequences of Theorem 3.1. 3.4.1 Totally transcendental ℵ0-categorial theories. First we dis... |

4 | Model theory of modules over a serial ring - Eklof, Herzog - 1995 |

3 | On the number of sets definable by polynomials
- Jerónimo, Sabia
(Show Context)
Citation Context ...r paper (see [6, Theorem 1.1]) we will show that the shatter function of any partitioned L-formula with m parameter variables (such as ϕ∗) is O(tm) in Th(K); hence π∗ϕ(t) = πϕ∗(t) = O(t m). (In fact, =-=[46]-=- proves that π∗ϕ(t) ≤ ∑m k=0 ( t k ) dk for every t, and this bound is asymptotically optimal.) 1.5. Organization of the paper. In the preliminary Section 2 we set the scene by recalling the definitio... |

3 |
Theories d’arbres.
- Parigot
- 1982
(Show Context)
Citation Context ...= (T,<) with the property that for each t ∈ T the set {t′ ∈ T : t′ < t} is linearly ordered (by the restriction of <). Problem. Determine the VC density function of each (infinite) tree. (It is known =-=[75]-=- that a tree T is stable iff T has finite height, and then T is superstable of U-rank ≤ height(T ), so conceivably, the methods of [6] could be applied.) 7. A Strengthening of VC d, and P -adic Exampl... |

3 | Essentially periodic ordered groups - Point, Wagner |

3 | groupes ω-stables de rang fini - Lascar - 1985 |

2 |
Types dans les corps valués
- Bélair
- 1999
(Show Context)
Citation Context ...elds 4 ASCHENBRENNER, DOLICH, HASKELL, MACPHERSON, AND STARCHENKO of characteristic (0, 0) and algebraically maximal Kaplansky fields of characteristic (p, p) are NIP iff their residue fields are NIP =-=[13, 12]-=-. On the other hand, each pseudofinite field (infinite model of the theory of all finite fields) is not NIP [29], since it defines the (Rado) random graph. 1.4. Uniform bounds on VC density. This pape... |

2 | Central Limit Theorems, Cambridge - Uniform - 1999 |

2 |
On uniform definability of types over finite sets, preprint (2010), available online at http://front.math.ucdavis.edu/1005.4924
- Guingona
(Show Context)
Citation Context ...v), we denote by F(y; c) the family (ψ(y; c))ψ∈F of L(M)-formulas obtained by substituting a given tuple c ∈ M |v| for the tuple of variables v. The following generalizes a definition due to Guingona =-=[34]-=-: Definition 5.1. We say that ∆ has uniform definability of types over finite sets (abbreviated as UDTFS ) in M if there are finitely many families Fi = ( ϕi(y; y1, . . . , yd) ) ϕ∈∆ (i ∈ I) of L-form... |

2 |
Local dp-rank and vc-density over indiscernible sequences
- Guingona, Hill
- 2011
(Show Context)
Citation Context ... next section shows. We do not know the answer to the following question: Question. Is vcind(ϕ) always integral-valued? (After a first version of this manuscript had been completed, Guingona and Hill =-=[35]-=- showed that this question indeed has a positive answer.) We finish this section with a connection between vc∗ind and the Helly number. We already remarked (see Section 2.4) that if M = (M,<) is a den... |

1 |
Maximum number of edges joining vertices on a cube
- Abdel-Ghaffar
(Show Context)
Citation Context ... t log t (1 + o(1)) as t→∞. VC DENSITY IN SOME NIP THEORIES, I 35 Proof. Set Ed(t) := max {|E[A]| : A ⊆ Qd, |A| = t} for d > 0 and t ≤ 2d. Then πϕ̂(t) = t + maxd≥⌈log t⌉Ed(t). It is known (see, e.g., =-=[1]-=-) that there is some function g with g(t) = 12 t log t (1 + o(1)) as t→ ∞ such that Ed(t) = g(t) for all d and t ≤ 2d. This yields the claim. 4.4.2. An example in R = (R, 0, 1,+,−,×, <). For this we... |

1 | Densité et dimension, Ann - Assouad - 1983 |

1 |
sur les classes de Vapnik-Cervonenkis et la dimension combinatoire de Blei, in: Seminaire d’Analyse Harmonique
- Observations
- 1983
(Show Context)
Citation Context ...) by Lemma 2.1, and vc(S) <∞ iff VC(S) <∞. The VC density of S is also known as the real density [7] or the VC exponent [17] of S. It is related to the combinatorial dimension of S introduced by Blei =-=[8]-=- and to compression schemes for S [47]. Example. Suppose S = (X≤d). Then the inequality in the statement of Lemma 2.1 is an equality, and VC(S) = vc(S) = d. Example. Suppose X = Rd, and S is the colle... |

1 |
Externally definable sets and dependent pairs, preprint (2010), available online at http://front.math.ucdavis.edu/1007.4468
- Chernikov, Simon
(Show Context)
Citation Context ...h expansion of M is the expansion of M to an LSh-structure MSh where each predicate symbol Rψ,c(x) as before is interpreted by M |x| ∩ ψM∗((M∗)|x|; c). Shelah showed [89] (with another proof given in =-=[20]-=-) that if T is NIP then T Sh = Th(MSh) admits 24 ASCHENBRENNER, DOLICH, HASKELL, MACPHERSON, AND STARCHENKO quantifier elimination and is also NIP. This provides an interesting way of constructing new... |

1 |
The homeomorphic embedding of Kn in the m-cube
- Hartman
- 1976
(Show Context)
Citation Context ...mbeds into G. Note that the graph Q is not superflat: in fact, for every d there is an embedding of a subdivision of Kd+1, obtained by placing at most one additional vertex on each edge, into Qd, cf. =-=[37]-=-. However, we do have: 34 ASCHENBRENNER, DOLICH, HASKELL, MACPHERSON, AND STARCHENKO Proposition 4.10. Q is ω-stable. Towards a proof of this proposition, we first introduce some notation and terminol... |

1 |
density in real closed valued fields, Prépublications de la séminaire de structures algébriques ordonnées 83 (2008–2009), Equipe de logique
- VC
(Show Context)
Citation Context ...vex valuation ring. In fact, the methods of Karpinski and Macintyre can also be adapted to give the correct density bounds for this and certain other weakly o-minimal expansions of real closed fields =-=[40]-=-. Some interesting weakly o-minimal theories to which these methods do not readily adapt may be found in [5, 54]. VC DENSITY IN SOME NIP THEORIES, I 5 Our approach to Theorem 1.1, via definable types,... |

1 |
Additivity of the dp-rank, preprint (2011), available online as no. 251 at http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server
- Kaplan, Onshuus, et al.
(Show Context)
Citation Context ...c∗(ϕ) ≥ 2. In particular, each of the theories in Examples 3.8–3.10 (being VC-minimal) is dpminimal. Other proofs of the dp-minimality of weakly o-minimal theories can be found in [2, 24]. See also =-=[48]-=- for a generalization of Corollary 5.13 to a bound on “dp-rank” in terms of VC density. A characterization of dp-minimal theories among stable theories was given in [71]. The main result of [34] is th... |

1 |
On the independence property
- Kudăıbergenov
(Show Context)
Citation Context ...rty if some L-formula has the independence property for M , and not to have the independence property (or to be NIP or dependent) otherwise. By a classical result of Shelah [86] (with other proofs in =-=[52, 55, 80]-=-), for M to be NIP it is actually sufficient that no formula ϕ(x; y) with |x| = 1 has the independence property for M . NIP is implied by (but not equivalent to) another prominent tameness condition o... |

1 |
quasi-o-minimal models, Siberian Adv
- Weakly
(Show Context)
Citation Context ... the corresponding expansion of (Z, <,+) has quantifier elimination in a natural expansion of {<,+, U} (see [19, 78]) and is NIP (as will be shown elsewhere). 6.3. Weakly quasi-o-minimal theories. In =-=[53]-=-, T is called weakly quasi-o-minimal if for any M |= T , any definable subset of M is a finite Boolean combination of convex subsets of M and ∅-definable sets. Every weakly quasi-o-minimal theory is N... |

1 |
On definable subets of p-adic fields
- Macintyre
- 1976
(Show Context)
Citation Context ... Here and below, p is a fixed prime number. Let pCF denote the Lp-theory of Qp, where each Pn is interpreted as the set of nth powers in Qp: Qp |= ∀x ( Pn(x)↔ ∃y(yn = x) ) . By a theorem of Macintyre =-=[60]-=-, pCF has elimination of quantifiers. Following [39], an L-theory T containing pCF is called P -minimal if, in every model of T , every definable subset in one variable is quantifier-free definable ju... |

1 |
VC-dimension implies a fractional Helly theorem, Discrete Comput
- Bounded
(Show Context)
Citation Context ...nibles: Corollary 3.26. Suppose the set system Sϕ is d-consistent, where d = ⌊vc∗(ϕ)⌋ + 1. Then there is an infinite subset of Sϕ which is consistent. This is a weak version of a theorem of Matoušek =-=[67]-=-, according to which, if Sϕ is d-consistent, where d > vc∗(ϕ), then one may write Sϕ = S1∪· · ·∪SN (for some N ∈ N) where each Si is consistent. 4. Some VC Density Calculations In this section we give... |

1 | 0-categorical distributive lattices of finite breadth - Schmerl - 1983 |

1 |
ordered sets and the independence property
- Partially
- 1989
(Show Context)
Citation Context ...1 property. This observation can be used to strengthen [92, Proposition 4.2], where it is shown that the complete theories of colored linearly ordered sets with monotone relations (shown to be NIP in =-=[85]-=-) are dp-minimal. A binary relation R on a set X is said to be monotone with respect to a linear ordering < of X if x′ ≤ xRy ≤ y′ ⇒ x′Ry′ for all x, x′, y, y′ ∈ X . A colored linearly ordered set with... |

1 |
dependent theories
- Strongly
(Show Context)
Citation Context ...y| such that for all i and j the set of L(M)-formulas{ α(x; ai), β(x; bj) } ∪ {¬α(x; ak) : k 6= i} ∪ {¬β(x; bl) : l 6= j} is consistent (with M). This notion and the following definition originate in =-=[90]-=-: Definition 5.11. An L-theory T is said to be dp-minimal if in no model of T there is an ICT pattern, and M is dp-mininmal if Th(M) is dp-minimal (equivalently, if there is no ICT pattern in an eleme... |

1 | On stability and products - Wierzejewski |

1 |
Magidor-Malitz quantifiers in modules
- Baudisch
- 1984
(Show Context)
Citation Context ...h at most dm. Thus by Proposition 4.10 we get vcT (m) ≤ dm. Remark. Every complete theory of a module is dimensional [33, Corollary 6.21] and hence does not have the finite cover property (see also =-=[4]-=-); hence Corollary 3.8 yields another proof of Corollary 4.24. The relationship between the ordered sets P̃Pm(M) and PP 0 m(M) is particularly clean if T is totally transcendental: Lemma 4.25. If T is... |

1 |
Dp-minimal theories: basic facts and examples, Notre Dame
- Dolich, Goodrick, et al.
(Show Context)
Citation Context ...timately connected with a strengthening of the NIP concept, called dp-minimality, which has recently received attention through the work of Shelah [39], Onshuus-Usvyatsov [28], Dolich-Goodrick-Lippel =-=[9]-=-, and others. Before we state some of the main results of this paper, we briefly recall the relevant terminology. In the rest of this introduction, L is a first-order language, T is a complete L-theor... |

1 |
Une preuve par la théorie de la déviation d’un théorème de
- Poizat
- 1978
(Show Context)
Citation Context ... U(T ) ≤ MR(T ), the previous corollary applies in particular if T has finite Morley rank and does not have the finite cover property. The following folklore result (see, e.g., [29, Proposition B.1], =-=[31]-=-, or [34, Theorem 4.5]) can be used to verify that T has finite Morley rank: Proposition 3.9. Suppose there is a family {Di}i∈I of strongly minimal definable sets such that every non-algebraic type in... |