General Non-structure Theory

The theme of the first two sections, is to prepare the framework of how from a " complicated " family of index models I ∈ K 1 we build many and/or complicated structures in a class K 2. The index models are characteristically linear orders, trees with κ + 1 levels (possibly with linear order on the set of successors of a member) and linearly ordered graph, for this we phrase relevant complicatedness properties (called bigness). We say when M ∈ K 2 is represented in I ∈ K 1. We give sufficient conditions when {M I : I ∈ K 1 λ } is complicated where for each I ∈ K 1 λ we build M I ∈ K 2 (usually ∈ K 2 λ) represented in it and reflecting to some degree its structure (e.g. for I a linear order we can build a model of an unstable first order class reflecting the order). If we understand enough we can even build e.g. rigid members of K 2 λ. Note that we mention " stable " , " superstable " , but in a self contained way, using an equivalent definition which is useful here and explicitly given. We also frame the use of generalizations of Ramsey and Erdös-Rado theorems to get models in which any I from the relevant K 1 is reflected. We give in some detail how this may apply to the class of separable reduced Abelian ˙ p-group and how we get relevant models for ordered graphs (via forcing). In the third section we show stronger results concerning linear orders. If for each linear order I of cardinality λ > ℵ 0 we can attach a model M I ∈ K λ in which the linear order can be embedded such that for enough cuts of I, their being omitted is reflected in M I , then there are 2 λ non-isomorphic cases. But in the end of the second section we show how the results on trees with ω+1 levels gives results on linear ordered (not covered by §3), on trees with ω + 1 levels see [Sh:331]. To get more we prove explicitly more on such trees. Those will be enough for results in model theory of Banach space of Shelah-Usvyatsov [ShUs:928].