A complicated family of trees with omega + 1 levels

Abstract: Our aim is to prove that if T is a complete first order theory, which is not superstable (no knowledge on this notion is required), included in a theory T_1 then for any lambda > |T_1| there are 2^lambda models of T_1 such that for any two of them the tau(T)-reducts of one is not elementarily embeddable into the tau(T)-reduct of the other, thus completing the investigation of [Sh:a, Ch. VIII]. Note the difference with the case of unstable T: there lambda > |T_1| + aleph_0 suffices. By [Sh:E59] it suffices for every such lambda to find a complicated enough family of trees with omega + 1 levels of cardinality lambda. If lambda is regular this is done already in [Sh:c, Ch. VIII]. The proof here (in sections 1,2) go by dividing to cases, each with its own combinatorics. In particular we have to use guessing clubs which was discovered for this aim. In S.3 we consider strongly aleph_varepsilon-saturated models of stable T (so if you do not know stability better just ignore this). We also deal with separable reduced Abelian p-groups. We then deal with various improvements of the earlier combinatorial results.