On density of old sets in Prikry type extensions

Every set of ordinals of cardinality κ in a Prikry extension with a measure over κ contains an old set of arbitrary large cardinality below κ, and, actually, it can be split into countably many old sets. What about sets bigger cardinalities? Clearly, any set of ordinals in a forcing extension of a regular cardinality above the cardinality of the forcing used, contains an old set of the same cardinality. Still cardinals in the interval (κ, 2κ] remain. Here we would like to address this type of questions. 1 A situation under 2 = κ. Let us start with the following observation. Proposition 1.1 Suppose that 2 = κ. Let U be a normal ultrafilter over κ and PU be the Prikry forcing with U . Then in V PU there is a subset of κ without old subsets of the same size. Proof. Work in V . Pick a generating sequence 〈Aα | α < κ〉 for U such that for every α ≤ β < κ, Aβ ⊆∗ Aα . It is possible since 2 = κ and U is normal. Define an other generating sequence 〈Aα | α < κ〉 as follows. Let C ⊆ κ be a club such that |[αν , αν+1]| = κ, for every ν < κ, where {αν | ν < κ} is an increasing enumeration of C. We set Aαν = Aν . Pick a surjective map h : κ → [κ]. Let, for every ν < κ, gν : (αν , αν+1) ←→ κ. Now if β ∈ (αν , αν+1), for some ν < κ, then set Aβ = Aαν ∪ h(gν(β)). Clearly, 〈Aα | α < κ〉 is a generating sequence for U and for every α ≤ β < κ, Aβ ⊆∗ Aα. Now let G ⊆ PU be generic and {κn | n < ω} be the corresponding Prikry sequence. Set X = {α < κ | Aα ⊇ {κn | n < ω}}. A ⊆∗ B means that |A \B| < κ