Infinite combinatrics and its applications

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The main achievement of this project was that we could establish the theory of so called the weak Freese-Nation property (WFN) of partial orderings. The study of WFN began with Fuchino-Koppelberg-Shelah paper in 1996. In Fuchino-Soukup 1997 we then realized that WFN is highly set-theoretic in terms of independency results and consistency strength often involved in such independency. In the framework of our project we could solve all the open problems posed in the Fuchino-Soukup paper mentioned above, and several other problems which came to our mind as natural questions during this research. For some results we obtaind in which some consistency strength of large cardinals is involved, exact equiconsistency remained to be determined. Though this problem seems to be very difficult if not possible, since the large cardinals employed in some of these concistency results are so large that there is no inner model theory known for them.One of the knowledge we obtained through the study of WFN … More was that the assumption of WFN of (P(ω)), ⊆) captures a lot of the combinatorial properties of Cohen models so that this assumption could be regarded as one of the natural axioms for Cohen models. Quite recently Istvan Juhasz and Kenneth Kunen introduced the property which they called SEP which generalizes both WFN and CH* of Juhasz, Soukup and Szentmiklossy. The last principle was also known as a natural axiom valid in Cohen models. In the new light of SEP, Juhasz and Kunen could obtain some very interesting results on WFN and CH*, and on their relations to CィイD1sィエD1(κ) of Juhasz, Soukup and Szentmiklossy, and to HP(κ) of D Fuchino and Brendle. It seems that this new setting with SEP and its variations offers a bunch of new problems and gives possibilities to recast known results on the principles mentioned above which were studied sofar rather separately to each other, so that they can be gradually put together to build a unified theory.In the framework of the project we could also obtain some other results in Pκ (λ)-combinatorics, set-theoretic analysis and set theory of reals as well. Less