Application of the Axiomatic Set Theory to the Infinitary Combinatorics

这个研究项目的主要课题是：（1）研究断言P（ω）具有弱自由民族性质（WFN）的原理及其变化;（2）研究连续统具有一些大基数性质（如实值可测性（RVM）或其范畴对偶）的集合论模型;（3）其他集合论或组合学课题，包括来自实数集合论的问题。关于（1），Fuchino和Stefan。Geschke可以得到关于SEP性质的一些基本结果，SEP性质是I.Juhasz和K. Kunen引入的P（ω）的WFN的一个弱化。在S.Shelah最近的一个结果的基础上，Fuchino可以重建Juhasz，Soukup，Szentmiklossy关于C^s（κ）原理的一个定理，并证明了这个原理与P（ω）的WFN的变化之间的联系。对于（2），Puchino和S.Shelah对大卫Fremlin提出的一个问题给出了肯定的回答，即是否存在一个RVM模型，它不同于Solovey的RVM标准模型，具有某种“数学”性质。关于（3），Fuchino与Stefan Geschke和Lajos Soukup研究了与ω的几乎不相交子集族有关的一类新的基数不变量，并得到了关于这些基数不变量的一些相容性结果。

The main subjects of this research project were:(1)study of the principle asserting that P(ω) has the weak Freese-Nation property (WFN), and its variations;(2)study of the models of set theory where the continuum has some large cardinal property like real-valued measurability (RVM) or its categorical dual;(3)other set-theoretic or combinatoric subjects including problems from the set-theory of reals.Concerning (1), Fuchino and Stefan. Geschke could obtain some fundamental results conserning the property SEP which is a weakening of WFN of P(ω) introduced by I.Juhasz and K.Kunen. Building on a recent result by S.Shelah, Fuchino could reconstruct a theorem by Juhasz, Soukup, Szentmiklossy on the principle C^s(κ) and showed the connections of this principle with variations of WFN of P(ω).For (2), Puchino and S.Shelah gave a positive answer to a question by David Fremlin whether there is a model of RVM which is different from Solovey's standard model of RVM by some "mathematical" property. The model found Fuchino and Shelah satisfies the club principle for some cardinal -κ, which is not the case for the standard model by Solovey.Concerning (3), Fuchino together with Stefan Geschke and Lajos Soukup studied some new type of cardinal invariants connected with almost disjoint family of subsets of ω and could obtain some consistency results on these cardinal invariants.