Paper detail

Ranks for strongly dependent theories

There is much more known about the family of superstable theories when compared to stable theories. This calls for a search of an analogous "super-dependent" characterization in the context of dependent theories. This problem has been treated in \cite{Sh:783,Sh:863}, where the candidates "Strongly dependent", "Strongly dependent^2" and others were considered. These families generated new families when we are considering intersections with the stable family. Here, continuing \cite[§2, §5E,F,G]{Sh:863}, we deal with several candidates, defined using dividing properties and related ranks of types. Those candidates are subfamilies of "Strongly dependent". Fulfilling some promises from \cite{Sh:863} in particular \cite[1.4(4)]{Sh:863}, we try to make this self contained within reason by repeating some things from there. More specifically we fulfil some promises from \cite{Sh:863} to to give more details, in particular: in \S4 for \cite[1.4(4)]{Sh:863}, in \S2 for \cite[5.47(2)=Ldw5.35(2)]{Sh:863} and in \S1 for \cite[5.49(2)]{Sh:863}