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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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Itay Kaplan , Alf Onshuus , and Alex Usvyatsov , Additivity of the dp-rank. Transactions of the American Mathematical Society, vol. 365 (2013), pp. 57835804.

Jirí Matousek , Bounded vc-dimension implies a fractional helly theorem. Discrete & Computational Geometry, vol. 31 (2004), pp. 251255, 10.1007/s00454-003-2859-z.

Pierre Simon , Distal and non-distal theories. Annals of Pure and Applied Logic, vol. 164 (2013), no. 3, pp. 294318.

K. Tent and M. Ziegler , 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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