Externally definable sets and dependent pairs

Abstract

We prove that externally definable sets in first order NIP theories have honest definitions, giving a new proof of Shelah's expansion theorem. Also we discuss a weak notion of stable embeddedness true in this context. Those results are then used to prove a general theorem on dependent pairs, which in particular answers a question of Baldwin and Benedikt on naming an indiscernible sequence.Comment: 17 pages, some typos and mistakes corrected, overall presentation improved, more details for the examples are give