What are some of the common popular stable theories that are known to be dp-minimal (or not dp-minimal)?


Some dp-minimal examples I am aware of are strongly minimal theories, superstable theories of U-rank 1, and infinitely many refining equivalence relations.


The particular examples I am interested in are:


$DCF\_0$


$DCF\_p$


free group on $n>1$ generators


everywhere infinite forest


In general I would also like to know (relatively) classical examples of $\omega$-stable, superstable, and strictly stable theories that are not dp-minimal. 



dp-minimality and stability
A stable theory is dp-minimal if and only if all 1-types have weight 1. (See "On dp-minimality, strong dependence and weight" by A. Onshuus and A. Usvyatsov.)


A differentially closed field has 1-types of arbitrary large finite weight, hence neither $DCF\_0$ nor $DCF\_p$ are dp-minimal. Similarly, the generic type of the free group has infinite weight---see Pillay's "On genericity and weight in the free group"---so it is not dp-minimal. (Alternatively, one can use the result I prove in "On dp-minimal ordered structure" that a dp-minimal group is abelian-by-finite exponent.)


As far as I see, the everywhere infinite forest is dp-minimal; this should not be hard to check by inspection.


This also answers your last question asking for non-dp-minimal stable structures. Other examples are separably closed fields and cross-cutting equivalence relations.


