. Then, ) are equivalent and imply (1) If T is NIP, then they are all equivalent

. Proof, This is enough Assume A = a i : i < ?, B = b i : i < ? and c are a witness to ¬(3) Then there are two tuples (i 1 < ... < i n ), (j 1 < ... < j n ) and a formula ?(x; y 1 , ..., y n ) such that | = ?(c; a i 1 , ..., a in ) ? ¬?(c; a j 1 , ..., a jn ) Take an ? < ? greater than all the i k and the j k . Then, exchanging the i k and j k if necessary, we may assume that | = ?(c; a i 1 , ..., a in ) ? ¬?(c; a n.? , ..., a n.?+n?1 ). Define §6.4 Dp-minimal groups We study inp-minimal groups. Note that by an example of Simonetta, ([64]), not all such groups are abelian-by-finite. It is proven in [39] that C-minimal groups are abelian-by-torsion, We generalize the statement here to all inp-minimal theories

. Proof and A. Let, To see this A for all n > 0. Then, for n = m, the cosets a m B and a n B are distinct, as are A.b m and A.b n . Now we obtain an independence pattern of length two by considering the sequences of formulas ? k (x) = ''x ? a k B " and ? k (x) = ''x ? A.b k " . For x ? G, let C(x) be the centralizer of x. By compactness, there is k such that for x, y ? G, for some k ? k, either x k ? C(y) or y k ? C(x) In particular, letting n = k!, x n and y n commute n )), the bicommutant of the nth powers of G. It is an abelian definable subgroup of G and for all x ? G, x n ? H. Finally, if H contains all n powers then it is also the case of all conjugates of H, so replacing H by the intersection of its conjugates, we obtain what we want Now we work with ordered groups, Note that in such a group, the convex hull of a subgroup is again a subgroup

. Proof, We may assume that H and C are ?-definable. So without loss, assume C = G. If H is not of finite index, there is a coset of H that forks over ?

?. One-color, C. Such, and M. , ? For each k, monotone relations R k and S k such that M | = x 0

U. Define, {t : tR k x k } and V k (x) = {t : xS k t}. The U k (x) are initial segments of M and the V k (x) final segments. For each k, k , either U k (x k ) ? U k (x k ) or U k

. Proof, By the previous result, we may assume that T = Th(M) eliminates quantifiers

. Proof, We can define such a structure in a linear order with monotone relations : see [52] More precisely, there exists a structure P = (P, R j ) in which is a linear order and the R j are monotone relations, and there is a definable relation O(x, y) such that the structure (P, O) is isomorphic to (M, ?) The result therefore follows from the previous one

. Proof, A straightforward back-and-forth, noticing that tp(¯ x/a)?tp(¯ y/a) tp qf (¯ x?¯ y/a) (quantifier-free type)

H. Adler, Strong theories, burden, and weight

H. Adler, An introduction to theories without the independence property, Archive of Mathematical Logic, 2008.

Y. Baisalov and B. Poizat, Paires de structures O-minimales, The Journal of Symbolic Logic, vol.59, issue.02, pp.570-578, 1998.
DOI : 10.1090/S0002-9947-1988-0943306-9

B. Baizhanov and J. Baldwin, Abstract., The Journal of Symbolic Logic, vol.26, issue.04, pp.1243-1260, 2004.
DOI : 10.2178/jsl/1102022221

J. Baldwin and M. Benedikt, Stability theory, permutations of indiscernibles , and embedded finite models. Transactions of the, pp.4937-4969, 2000.

O. Belegradek, F. Peterzil, and . Wagner, Abstract, The Journal of Symbolic Logic, vol.77, issue.03, pp.1115-1132, 2000.
DOI : 10.2307/2586690

O. Belegradek and B. Zilber, The model theory of the field of reals with a subgroup of the unit circle, Journal of the London Mathematical Society, vol.78, issue.3, pp.78563-579, 2008.
DOI : 10.1112/jlms/jdn037

A. Berenstein, Lovely pairs and dense pairs of o-minimal structures

A. Berenstein, A. Dolich, and A. Onshuus, Abstract, The Journal of Symbolic Logic, vol.71, issue.02, pp.391-404, 2011.
DOI : 10.1016/S0168-0072(02)00060-X

A. Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth, Learnability and the Vapnik-Chervonenkis dimension, Journal of the ACM, vol.36, issue.4, pp.929-965, 1989.
DOI : 10.1145/76359.76371

G. Boxall, Nip for some pair-like theories Archive for Mathematical Logic, pp.353-359, 2011.

E. Casanovas and M. Ziegler, Stable theories with a new predicate, The Journal of Symbolic Logic, vol.53, issue.03, pp.1127-1140, 2001.
DOI : 10.1090/S0002-9947-00-02672-6

Z. Chatzidakis and E. Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc, vol.351, issue.8, pp.2997-3071, 1999.
DOI : 10.1017/9781316755907.003

A. Chenikov, Theories without the tree property of the second kind

A. Chernikov and I. Kaplan, Forking and dividing in NTP 2 theories, 2009.

A. Chernikov and P. Simon, Externally definable sets and dependent pairs, Israel Journal of Mathematics, vol.173, issue.1
DOI : 10.1007/s11856-012-0061-9

URL : https://hal.archives-ouvertes.fr/hal-00526000

F. Delon, Definissabilite Avec Parametres Exterieurs Dans Q p Et R, Proceedings of the American Mathematical Society, vol.106, issue.1, pp.193-198, 1989.
DOI : 10.2307/2047391

A. Dolich, D. Lippel, and J. Goodrick, Dp-minimal theories : basic facts and examples
DOI : 10.1215/00294527-1435456

URL : http://arxiv.org/abs/0910.3189

J. Antonio, A. Engler, and . Prestel, Valued fields, 2005.

A. Fornasiero, Tame structures and open cores, 2010.

J. Goodrick, Abstract, The Journal of Symbolic Logic, vol.162, issue.01, pp.221-238, 2010.
DOI : 10.1090/S0002-9947-00-02633-7

V. Guingona, Dependence and isolated extensions. Modnet preprint 212, 2009.

V. Guingona, On uniform definability of types over finite sets. Modnet preprint 250, 2010.

A. Günaydin and P. Hieronymi, The real field with the rational points of an elliptic curve

B. Hart, B. Kim, and A. Pillay, Coordinatisation and canonical bases in simple theories, The Journal of Symbolic Logic, vol.65, issue.01, pp.293-309, 2000.
DOI : 10.1007/BF02760649

D. Haskell, E. Hrushovski, and D. Macpherson, Stable domination and independence in algebraically closed valued fields, volume 30 of Lecture Notes in Logic, 2008.

W. Hodges, Model Theory, volume 42 of Encyclopedia of mathematics and its applications, Great Britain, 1993.

E. Hrushovsk, A. Pillay, and P. Simon, Generically stable and smooth measures in NIP theories, Transactions of the American Mathematical Society, vol.365, issue.5
DOI : 10.1090/S0002-9947-2012-05626-1

E. Hrushovski, Pseudo-finite fields and related structures, Model theory and applications, pp.151-212, 2002.

E. Hrushovski and F. Loeser, Non-archimedean tame topology and stably dominated types, 2010.
DOI : 10.1515/9781400881222

URL : http://arxiv.org/abs/1009.0252

E. Hrushovski, A. Peterzil, and . Pillay, Groups, measures, and the NIP, Journal of the American Mathematical Society, vol.21, issue.02, pp.563-596, 2008.
DOI : 10.1090/S0894-0347-07-00558-9

E. Hrushovski and A. Pillay, On NIP and invariant measures, Journal of the European Mathematical Society, vol.13, pp.1005-1061, 2011.
DOI : 10.4171/JEMS/274

URL : http://arxiv.org/abs/0710.2330

H. R. Johnson and M. C. Laskowski, Compression Schemes, Stable Definable Families, and??o-Minimal Structures, Discrete & Computational Geometry, vol.5, issue.4, pp.914-926, 2010.
DOI : 10.1007/s00454-009-9201-3

M. Karpinski and A. Macintyre, Approximating volumes and integrals in o-minimal and p-minimal theories In Connections between model theory and algebraic and analytic geometry, Quad. Mat, vol.6, pp.149-177, 2000.

H. and J. Keisler, Choosing elements in a saturated model, Classification theory, pp.165-181, 1985.
DOI : 10.1007/BFb0082237

H. and J. Keisler, Measures and forking, Annals of Pure and Applied Logic, vol.34, issue.2, pp.119-169, 1987.
DOI : 10.1016/0168-0072(87)90069-8

B. Kim and A. Pillay, Simple theories, Annals of Pure and Applied Logic, vol.88, issue.2-3, pp.149-164, 1995.
DOI : 10.1016/S0168-0072(97)00019-5

D. Macpherson and C. Steinhorn, On variants of o-minimality, Annals of Pure and Applied Logic, vol.79, issue.2, pp.165-209, 1996.
DOI : 10.1016/0168-0072(95)00037-2

D. Marker and C. I. Steinhorn, Definable types in -minimal theories, The Journal of Symbolic Logic, vol.52, issue.01, pp.185-198, 1994.
DOI : 10.1090/S0002-9947-1988-0943306-9

J. Matousek, Lectures on Discrete Geometry, 2002.
DOI : 10.1007/978-1-4613-0039-7

M. Morley, Categoricity in power. Transaction of the, pp.514-538, 1965.

A. Onshuus and A. Pillay, -adic Lie groups, Journal of the London Mathematical Society, vol.78, issue.1, pp.233-247, 2008.
DOI : 10.1112/jlms/jdn018

URL : https://hal.archives-ouvertes.fr/hal-01211267

A. Onshuus and Y. Peterzil, A note on stable sets, groups, and theories with NIP, MLQ, vol.59, issue.3, 2007.
DOI : 10.1002/malq.200610046

A. Onshuus and A. Usvyatsov, Abstract, The Journal of Symbolic Logic, vol.44, issue.03
DOI : 10.1007/s11856-009-0082-1

M. Parigot, Th??ories d'arbres, The Journal of Symbolic Logic, vol.46, issue.04, pp.841-853, 1982.
DOI : 10.1007/BF02757234

A. Pillay, Geometric stability theory. Oxford logic guides, 1996.

A. Pillay, On externally definable sets and a theorem of shelah. Algebra, logic, set theory, Stud, Log. (Lond.), vol.4, pp.175-181, 2007.

B. Poizat, Paires de structures stables, The Journal of Symbolic Logic, vol.44, issue.02, pp.239-249, 1983.
DOI : 10.1016/0003-4843(71)90015-5

B. Poizat, Cours de théorie des modèles. Nur al-Mantiq wal-Mari'fah, 1985.

M. Rubin, Theories of linear order, Israel Journal of Mathematics, vol.64, issue.4, pp.392-443, 1974.
DOI : 10.1007/BF02757141

J. H. Schmerl, Abstract, The Journal of Symbolic Logic, vol.141, issue.02, pp.396-401, 1989.
DOI : 10.1016/0003-4843(71)90015-5

S. Shelah, Classification theory and the number of nonisomorphic models, volume 92 of Studies in Logic and the Foundations of Mathematics, 1990.

S. Shelah, Classification theory for elementary classes with the dependence property?a modest beginning, Sci. Math. Jpn, vol.59, issue.2, pp.265-316, 2004.

S. Shelah, Dependent first order theories, continued, Israel Journal of Mathematics, vol.136, issue.2, pp.1-60, 2009.
DOI : 10.1007/s11856-009-0082-1

URL : http://arxiv.org/abs/math/0406440

S. Shelah, A dependent dream and recounting types. preprint, 2010.

S. Shelah, Dependent theories and the generic pair conjecture, Communications in Contemporary Mathematics, vol.17, issue.01, 2010.
DOI : 10.1142/S0219199715500042

S. Shelah and P. Simon, Adding linear orders. Submitted, Modnet preprint 290, 2011.

P. Simon, Théories nip. Master's thesis, École normale supérieure, 2009.

P. Simon, Distal and non-distal NIP theories, Annals of Pure and Applied Logic, vol.164, issue.3, 2011.
DOI : 10.1016/j.apal.2012.10.015

P. Simon, Finding generically stable measures. to appear in the Journal of Symbolic Logic, 2011.
DOI : 10.2178/jsl/1327068702

URL : http://arxiv.org/abs/1009.3566

P. Simon, Abstract, The Journal of Symbolic Logic, vol.54, issue.02, pp.448-460, 2011.
DOI : 10.1016/0168-0072(95)00037-2

URL : https://hal.archives-ouvertes.fr/inria-00075065

P. Simonetta, An example of a $C$-minimal group which is not abelian-by-finite, Proceedings of the American Mathematical Society, vol.131, issue.12
DOI : 10.1090/S0002-9939-03-06969-7

A. Usvyatsov, Abstract, The Journal of Symbolic Logic, vol.59, issue.01
DOI : 10.1305/ndjfl/1040248458

L. Van-den and . Dries, Dense pairs of o-minimal structures, Fund. Math, vol.157, pp.61-78, 1998.

L. Van-den-dries and A. H. Lewenberg, T-convexity and tame extensions, Journal of Symbolic Logic, vol.155, issue.3, pp.807-836, 1995.

V. N. Vapnik, A. Ya, and . Chervonenkis, On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities, Theory of Probability & Its Applications, vol.16, issue.2, pp.264-280, 1971.
DOI : 10.1137/1116025

O. Frank and . Wagner, Simple theories, volume 503 of Mathematics and its Applications, 2000.

I. , B. Yaacov, and H. J. Keisler, Randomization of models as metric structures, Confluentes Mathematici, vol.1, issue.2, pp.197-223, 2009.

Y. Itaï-ben, Continuous and random vapnik-chervonenkis classes, Israel Journal of Mathematics, vol.173, pp.309-333, 2009.