Few Non Minimal Types and Non-structure Sh603

We deal with abstract elementary classes K which has amalgamation in λ, categorical in λ and in λ. Our main result is that if (iω ≤ 2 < 2 + and) the minimal types on members of Kλ are not dense (among non algebraic (complete) types over models in Kλ, extending our given model), then the number of models in K of cardinality λ is maximal. For this we deal with some claims in pcf. This improves a result in [Sh 576], it is essentially self-contained. Saharon: (a) in 3.4 proof: |S∗(M)| ≥ limκ(T ), reference (but 3.4 contains a proof, 1.15 contains a statement) (b) also in 1.12; there is fact in end of §1 (c) weaker coding: 3.4 refer to Definition ? 2000 Mathematics Subject Classification. 03C45, 03C75, 03C95, 03E04.