Characterizing forking in VC-minimal theories

We consider the class of VC-minimal theories, as introduced by Adler in [2]. After covering some basic results, including a notion of generic types, we consider two kinds of VC-minimal theories: those whose generating directed families are unpackable and almost unpackable. We introduce two new decompositions of definable sets in VC-minimal theories, the layer decomposition and the irreducible decomposition, which allow for more precision than the standard Swiss cheese decompositon with regard to parameters. Finally, after introducing a slight generalization of the classical notions of forking and dividing, we prove that in any VC-minimal or quasi-VC-minimal theory whose generating family is unpackable or almost unpackable, forking of formulae over a model M is equivalent to containment in a global M -definable type, generalizing a result of Dolich on o-minimal theories in [8].