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This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality.

In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimension coming from the order.

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[1]Chernikov, Artem and Kaplan, Itay, Forking and dividing in NTP2theories, this Journal, vol. 77 (2012), no. 1, pp. 1–20.
[2]Chernikov, Artem and Simon, Pierre, Externally definable sets and dpendent pairs ii. Transactions of the American Mathematical Society, to appear.
[3]Cotter, Sarah and Starchenko, Sergei, Forking in vc-minimal theories, this Journal, vol. 77 (2012), no. 4, pp. 1257–1271.
[4]Kaplan, Itay, Onshuus, Alf, and Usvyatsov, Alex, Additivity of the dp-rank. Transactions of the American Mathematical Society, vol. 365 (2013), pp. 57835804.
[5]Kaplan, Itay and Shelah, Saharon, Examples in dependent theories, this Journal, 2012, to appear.
[6]Kaplan, Itay and Simon, Pierre, Witnessing dp-rank. Notre Dame Journal of Formal Logic, 2013, to appear.
[7]Matousek, Jirí, Bounded vc-dimension implies a fractional helly theorem. Discrete & Computational Geometry, vol. 31 (2004), pp. 251255, 10.1007/s00454-003-2859-z.
[8]Shelah, Saharon, Dependent theories and the generic pair conjecture. Communications in Contemporary Mathematics, vol. submitted (2010), math.LO/0702292.
[9]Simon, Pierre, On dp-minimal ordered structures, this Journal, vol. 76 (2011), no. 4, pp. 448–460.
[10]Simon, Pierre, Distal and non-distal theories. Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 294318.
[11]Simon, Pierre, A guide to NIP theories, Lecture Notes in Logic, 2014, to appear.
[12]Simon, Pierre and Starchenko, Sergei, On forking and definability of types in some dp-minimal theories, this Journal, vol. 79 (2014), no. 4, pp. 1020–1024.
[13]Tent, K. and Ziegler, M., A course in model theory, Lecture Notes in Logic, Cambridge University Press, 2012.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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