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 -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. (shrink)

We construct a nontrivial definable type V field topology on any dp-minimal field K that is not strongly minimal, and prove that definable subsets of Kn have small boundary. Using this topology and...

We survey the history of Shelah’s conjecture on strongly dependent fields, give an equivalent formulation in terms of a classification of strongly dependent fields and prove that the conjecture implies that every strongly dependent field has finite dp-rank.