Non-forking and preservation of NIP and dp-rank

We investigate the question of whether the restriction of an NIP type p is an element of S(B) which does not fork over A subset of Bto Ais also NIP, and the analogous question for dp-rank. We show that if Bcontains a Morley sequence Igenerated by pover A, then p vertical bar AIis 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. (C) 2021 Elsevier B.V. All rights reserved.