Ideals on Ρ_κλ and infinitary combinatorics related to large cardinal axiom

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(1)We proved any almost ineffable subset of Ρ_κλis ineffable provided that λ^<<κ>=2^λ.(2)We showed κ is not always λ^<<κ> ineffable even if it is λ ineffable.(3)The following sets were shown to have the weak partition property if Ρ_κλ has the property :(a){χ∈Ρ_κλ : χ∩κ is not an ordinal}, {χ∈Ρ_κλ : χ∩κ=|χ|}, and{χ∈Ρ_κλ : χ∩κ<|χ|}(b){χ∈Ρ_κλ : χ∩κ∈Α}where is Α any unbounded subset of κ.(4)We presented two forcing notions such that (a) and (b) holds in the generic extension respectively :(a)There exists a stationary subset of Ρ_κκ^+ which does not split into κ^+ many stationary sets.(b)There exists κ dense ideal on κ.These are simpler than known methods by Gitik for (a) and Woodin for (b).(5)We proved the following on stationary reflection (SR) :(a)If σ SR for Ρ_κλ holds and λ^2^<2<κ>=λ, then the σ club filter on Ρ_κκ is precipitous.(b)If κ SR for Ρ_<ω1>λ holds and λ^<2κ>=λ, then the club filter on κ has a weak covering property.(c)If κ SR for Ρ_<ω1>λ holds, λ^κ^+=λ, and 2^ω<κ<2^ω_1=^2<<κ>, then Ρ_κκ^+ splits into 2^ω_1 many stationary sets.(6)We built forcing models for (a)〜(c) respectively :(a)b=θ^*=cov(Μ), (b)d=θ^*ω_1 and θ=ω_1 and ω_2, (c)cov(Μ)=ω_1 and θ^*=θ=ω_2.(7)For partially orderd sets we define the property "SEP" to show :(a)non(Μ)=ω_1 when Ρ(ω) has SEP. If we further assume □ _<ω1>, then the minimal cardinality of almost disjoint family in Ρ(ω) is also ω_1.(b)We can build forcing models each of SEP for Ρ(ω) and -SEP for Ρ(ω).