Main Gap for Locally Saturated Elementary Submodels of A Homogeneous Structure

We prove a main gap theorem for locally saturated submodels of a homogeneous structure. We also study the number of locally saturated models, which are not elementarily embeddable to each other. Through out this paper we assume that M is a homogeneous model of similarity type (=language) L . We study elementary submodels of M . We use M as the monster model is used in stability theory and so we assume that the cardinality of M is large enough for all constructions we do in this paper. In fact, as in [HS1], we assume that |M| is strongly inaccessible. Alternatively we could assume less about |M| and instead of studying all elementary submodels of M , we could study suitably small ones. Also the assumption that M is homogeneous can be replaced by the assumption that M is κ-homogeneous for κ large enough. Notice that by [Sh1], if D is a stable finite diagram, then D has a monster model like M . We assume that the reader is familiar with [HS1] and use its notions and results freely. 0.1 Definition. (i) Suppose M is stable. We say that A is s-saturated if it is F λ(M) -saturated i.e. for all A ⊆ A of power < λ(M) and a there is b ∈ A such that t(b, A) = t(a,A) . (ii) We say that A is locally F κ -saturated if for all finite A ⊆ A there is F M κ -saturated B such that A ⊆ B ⊆ A . If M is stable, then we say that A is e -saturated if it is locally F λ(M) -saturated. (iii) Suppose M is stable. We say that A is strongly F κ -saturated if for all A ⊆ A of power < κ and a there is b ∈ A such that b E min,A a . By a-saturated we mean strongly F M κ(M) -saturated. ∗ Research supported by the United States-Israel Binational Science Foundation. Publ. 676.