Lecture 6.5 Saturation and homogeneity

The goal is to derive properties on embedding models from the realization of Galois types. We want to show that if M realizes ‘enough’ types over M then any small extension N of M can be embedded into M. The idea is first published as ‘saturation = model-homogeneity’ in 3.10 of [6] (Theorem 8 below), where the proof is incomplete. Successive expositions in [5, 2], and by Baldwin led to this version, where the key lemma was isolated by Kolesnikov. In contrast to various of the expositions and like Shelah, we make no amalgamation hypothesis.