Applications of pcf to ideals on P_κλ and infinitary combinatorics, and independence proof

(1)给出了当κ为λ超紧时，在P_κλ上存在正规超滤子的Kamo定理的一个简单证明。(2)我们证明了不具有弱分割性质的P_κλ ' a子集的理想不是κ^+饱和的。(3)给出了P_κλ上的最小理想，其划分性质不为λ^+饱和。(4)利用划分性质证明了P_κλ上的极小正规理想是非完全不可说子集理想的适当推广。(5)当ω_1<κ<λ时，P_κλ的平稳反射原理失效。(6)我们给出了一个强迫模型，其中对于每一个正则μ【小于或等于】κ， P_μκ上的非平稳理想对{x:cf(sup(x))=ω}的约束是弱预饱和的。(7)我们证明了在P_κ2^2^2^<<κ>的σ平稳反射下，P_κ上的非平稳理想对{x:cf(sup(x))=ω}的限制是弱饱和的。(8)给出了P_κλ的无界子集的最小基数等于平稳子集的最小基数的Shelah定理的一个正确的、简单的证明。(9)证明了当2^ω=2^<<κ>时，p <ω2>λ存在菱形序列，并给出了当2^ω<2^<ω_1>时，p <ω2>λ存在菱形序列的强迫模型。（10）证明了理想的可加性Ι_f)不大于有界数，且Ι_f)不小于支配数。（11）我们给出一个函数f∈^ωω，使得T_f不是理想。

(1)We give a simple proof for Kamo's theorem that there exists a normal ultrafilter on P_κλ if κ is λ supercompact.(2)We show the ideal of the subsets of P_κλ」a without the weak partition property is not κ^+ saturated.(3)We showed the minimal ideal on P_κλ with the partition property is not λ^+ saturated.(4)We proved the minimal normal ideal on P_κλ with the partition priperty is a proper extension of the ideal of noncompletely ineffable subsets.(5)We showed stationary reflection principle for P_κλ fails if ω_1<κ<λ.(6)We present a forcing model in which, for every regular μ【less than or equal】κ, the restriction of the nonstationary ideal on P_μκ to {x:cf(sup(x))=ω} is weakly presaturated.(7)We show that the restriction of the nonstationary ideal on P_κ to {x:cf(sup(x))=ω} is weakly presaturated under the σ stationary reflection in P_κ2^2^2^<<κ>.(8)We give a correct and simple proof for Shelah's theorem that the minimal cardinality of unbounded subset of P_κλ is equal to that of stationary subset.(9)We prove the existence of a diamond sequence for P_κλ provided that 2^ω=2^<<κ>, and present a forcing model such that P_<ω2>λ carries a diamond sequence whereas 2^ω<2^<ω_1>.(10)We showed that the additivity of the ideal Ι_f) is not, greater than the bounded number, and the cofinality of Ι_f) is not smaller than the dominating number.(11)We present a function f∈^ωω such that T_f is not an ideal.