On the Number of Elementary Submodels of an Unsuperstable Homogeneous Structure

We show that if M is a stable unsuperstable homogeneous structure, then for most κ < |M| , the number of elementary submodels of M of power κ is 2 . Through out this paper we assume that M is a stable unsuperstable homogeneous model such that |M| is strongly inaccessible (= regular and strong limit). We can drop this last assumption if instead of all elementary submodels of M we study only suitably small ones. Notice also that we do not assume that Th(M) is stable. We assume that the reader is familiar with [HS] and use all the notions and results of it freely. In [Hy1] a strong nonstructure theorem was proved for the elementary submodels of M assuming the existence of Skolem-functions. In this paper we drop the assumption on the Skolem-functions and prove the following nonstructure theorem. 1 Theorem. Let λ be the least regular cardinal ≥ λ(M) . Assume κ is an uncountable regular cardinal (< |M|) such that κ > λ and κ = κ . Then there are models (=elementary submodels of M) Ai , i < 2 , such that for all i < 2 , |Ai| = κ and for all i < j < 2 , Ai 6∼= Aj . See [Hy1] for nonstructure results in the case M is unstable. We prove Theorem 1 in a serie of lemmas. Let λ and κ be as in Theorem 1. By λ-saturated, λ-primary etc., we mean F λ -saturated, F M λ -primary etc. Notice that M is λ-stable. ∗ Research supported by the United States-Israel Binational Science Foundation. Publ. 632.