We give some sufficient conditions for a predicate P in a complete theory T to be “stably embedded”. Let 𝒫 be P with its “induced ∅-definable structure”. The conditions are that 𝒫 (or rather its theory) is “rosy”, P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus  which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict nonforking and weight in NIP theories (, ).
Anand Pillay. "Stable embeddedness and NIP." J. Symbolic Logic 76 (2) 665 - 672, June 2011. https://doi.org/10.2178/jsl/1305810769