Structures with pistes - different sizes

We generalize [5] and introduce structures with pistes which may have different number of models in every cardinality. 1 Structures with pistes–general setting. Assume GCH. As in [5], the first part (1.1) describes this ”linear” part of conditions in the main forcing. It is called a wide piste and incorporates together elementary chains of models of different cardinalities. The main forcing, defined in 1.2, will be based on such wide pistes and involves an additional natural but non-linear component called splitting or reflection. Definition 1.1 Let η < θ be regular cardinals, S be a function from the set {τ | η ≤ τ ≤ θ and τ is a regular cardinal } to θ, such that for every τ ∈ dom(S),S(τ) is a cardinal < τ . Assume also that S(τ) ≤ S(θ), for every τ ∈ dom(S). A (θ, η,S)−wide piste is a set 〈〈C , C〉 | τ ∈ s〉 such that the following hold. Let us first specify sizes of models that are involved. 1. (Support) s is a closed set of regular cardinals from the interval [η, θ] satisfying the following: (a) |s| < η, ∗The author was partially supported by Israel Science Foundation Grant No. 58/14. The requirement S(τ) < τ is essential for properness arguments. Once S(τ) = τ , then we are basically in a situation of [1] and arguments around chain condition and strategic closure replace properness. We need the ability to cover s by models of the least possible size, so s of cardinality above η is not allowed. Also, models of the least allowed cardinality η are not more than < η−closed, so it puts an additional restriction on the size of the support.