We investigate the question of whether the restriction of an NIP type p∈S(B) which does not fork over A⊆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↾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.
Bibliographical noteFunding Information:
The collaboration started during a research stay of the first author in The Hebrew University of Jerusalem partially supported by Fundaci?n Montcelimar, MTM2017-86777-P and 2017SGR-270.The second author would like to thank The Israel Science Foundation for their support of this research (Grants no. 1533/14 and 1254/18).
© 2021 Elsevier B.V.
- NIP/stable types