Application of the Axiomatic Set Theory to the Infinitary Combinatorics

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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.