Ranks for strongly dependent theories
Abstract
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 Sh:783,Sh:863, where the candidates "Strongly dependent", "Strongly dependent2" and others were considered. These families generated new families when we are considering intersections with the stable family. Here, continuing [ 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 Sh:863 in particular [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 Sh:863 to to give more details, in particular: in 4 for [1.4(4)]Sh:863, in 2 for [5.47(2)=Ldw5.35(2)]Sh:863 and in 1 for [5.49(2)]Sh:863