Applications of forcing and large cardinal axioms to infinitary combinatorics

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(1) By an iterated forcing we built a model of a supercompact cardinal in which for any F : [P_κλ]^2 → 2 every unbounded set in P_κλ has an unbounded homogeneous subset.(2) We proved P_κλ has the partition property if κ is λ ineffable and λ has cofinality greater than or equal to κ.(3) We showed there exists a normal ultrafilter on P_κλ with the partition property if κ is λ supercompact.(4) It was shown for any strongly normal κ saturated ideal I on P_κλ stationary reflection modulo I holds. While, by an iterated forcing, it was proved stationary reflection is not always true for strongly normal κ^+ saturated ideal.(5) We presented a forcing adding a stationary subset of P_κλ which does not reflect on a certain large set.(6) It was shown that P_κλ is subtle in a stronger sense than Menas had defined if κ is subtle.(7) By a forcing we gave a model with a real-valued measurable cardinal smaller than the cardinality of the continuum.(8) It was proved P_κλ has a non-reflecting stationary set if P_<κ^+>λ has an unbounded set of cardinality λ and P_νγ has an unbounded set of cardinality less than κ for any γ, ν less than κ.(9) We showed the strong club filter on P_κλ cannot be generated by the final segment filter and an unbounded set.(10) We proved the club filter is got from the strong club filter by the weak diagonal operation if λ has cofinality less than κ and the square principle holds for λ.