Non-forking and preservation of NIP and dp-rank

TL;DR

This paper explores how NIP and dp-rank properties are preserved under restrictions in model theory, providing conditions for preservation and counterexamples through new constructions.

Contribution

It establishes conditions under which NIP and dp-rank are preserved when restricting types, and introduces a novel construction of models demonstrating failure of preservation.

Findings

NIP and dp-rank are preserved when restricting to types containing a Morley sequence.

Positive results for generically stable NIP and stable types.

Counterexample using a new model construction showing non-preservation.

Abstract

We investigate the question of whether the restriction of a NIP type $p\in S(B)$ which does not fork over $A\subseteq B$ to $A$ is also NIP, and the analogous question for dp-rank. We show that if $B$ contains a Morley sequence $I$ generated by $p$ over $A$, then $p\restriction AI$ is NIP and similarly preserves the dp-rank. This yields positive answers for generically stable NIP types and the analogous case of stable types. With similar techniques we also provide a new more direct proof for the latter. Moreover, we introduce a general construction of "trees whose open cones are models of some theory" and in particular an inp-minimal theory DTR of dense trees with random graphs on open cones, which exemplifies a negative answer to the question.