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.