Applications of forcing and large cardinal axioms to infinitary combinatorics

(1)通过迭代强迫建立了一个超紧基数模型，其中对任意F：[P_κλ]^2→2，P_κλ中的每个无界集都有一个无界齐次子集。(2)我们证明了P_κλ具有分割性，如果κ是λ不可说的，并且λ具有大于或等于κ的余定性。(3)我们证明了如果κλ是κ超紧的，则P_λ上存在一个具有分割性的正规超滤子。(4)证明了对于模I的P_κ平稳反射上的任何强正规κλ饱和理想I都成立。同时，通过迭代强迫，证明了对于强正规κ^+饱和理想，平稳反射并不总是正确的。(5)给出了一种强迫加P_κλ的平稳子集，它不反射在某个大集上。(6)证明了当P_κλ是精细的时，P_κ具有比Menas所定义的更强意义上的精细。(7)通过强迫，我们给出了一个模型，它的实值可测基数小于连续统的基数。(8)证明了P_κλ存在非反射平稳集当P_&lt；κ^+&gt；λ有一个无界基数集λ，且P_νγ对任何小于κ的γ，ν都有一个小于κ的无界基数集。(9)我们证明了P_κλ上的强俱乐部滤子不能由最终段滤子和一个无界集合生成。(10)我们证明了如果λ有小于κ的余定性，并且λ的平方原理成立，则该俱乐部滤子是由强俱乐部滤子通过弱对角运算得到的。

(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.