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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.

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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.

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This comes a bit late. I just wanted to mention that an interesting family of examples are superflat graphs, proved to be stable by Podewski and Ziegler. See "Computations of Vapnik-Chervonenkis Density in Various Model-Theoretic Structures" by A. Bobkov. The graphs in the subclass of ultraflat graphs are omega stable.

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