Application of ideals for Godel's Program

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Let X be a subset of P_κλ. For each ordinal α【less than or equal】λ, let X_α = {s∈X:sup(s) = α}.We say that X is skinny if ∀α【less than or equal】λ(|X_α|<2^<|α|>) holds. Furthermore X is said to be skinnier if ∀α【less than or equal】λ(|X_α|<a) and skinniest if ∀α【less than or equal】λ(|X_α|【less than or equal】1). We prove the following :For a strong limit cardinal λ, a stationary subset X has a skinny stationary subset if NS_<κλ>|X is either precipitous or 2^λ saturated. On the other hand, Shelah proved that there is no skinny stationary subset of P_κλ if λ is a singular strong limit cardinal. Thus we obtained the following theorem :TheoremIf λ, is a singular strong limit cardinal, then NS_<κλ> is nowhere precipitous and nowhere 2^λ saturated.We proved that the existence of a skinnier stationary subset X of P_κλ, and λ^<<λ>=λ imply ◇_λ(E_X) where E_X = {sup(s):s∈X}. Clearly "skinny" and "skinnier" are equivalent under GCH. Therefore, assuming GCH ; if NS_<κλ> is precipitous or is 2^λ saturated, then ◇_λ(F) holds for every stationary subset F of E^λ_<<K>={α<λ:cf (α)<κ}.