Few non-minimal types and non-structure

Abstract

We pay two debts from [Sh:576] (math.LO/9805146). The main demands little knowledge from [Sh:576], just quoting a model theoretic consequence of the weak diamond. We assume that K has amalgamation in lambda, and that the minimal types are not dense to get many non-isomorphic models in lambda^+. For this also pcf considerations are relevant. The minor debt was the use in one point of [Sh:576] of lambda not= aleph_0, it is minor as for this case by [Sh:88] we ``usually'' know more.