Dp-minimality

A theory $$T$$ with home sort $$M$$ is said to be dp-minimal if it satisfies one of the following equivalent conditions:


 * Whenever $$I$$ and $$I'$$ are two mutually indiscernible sequences and $$a$$ is a singleton from the home sort, one of $$I$$ or $$I'$$ is indiscernible over $$a$$.
 * Whenever $$\langle b_i \rangle_{i \in I}$$ is an indiscernible sequence and $$a$$ is a singleton from the home sort, there is some $$i_0 \in I$$ such that $$\langle b_i \rangle_{i > i_0}$$ and $$\langle b_i \rangle_{i < i_0}$$ are $$a$$-indiscernible.
 * The home sort has dp-rank 1.

TODO: check that these definitions are correct.

If a theory is dp-minimal, it is NIP, and in fact strongly dependent. Strongly-minimal, o-minimal, C-minimal, and p-minimal theories are all dp-minimal, as are theories of VC-density 1. (TODO: check these claims.)

There is more generally a notion of dp-rank, and a set is dp-minimal if and only if it has dp-rank 1.

Expanding the theory by naming constants preserves dp-minimality.