Annals of Pure and Applied Logic 172 (1):102856 (2021)

We introduce the property “F-linked” of subsets of posets for a given free filter F on the natural numbers, and define the properties “μ-F-linked” and “θ-F-Knaster” for posets in a natural way. We show that θ-F-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct θ-Fr-Knaster posets (where Fr is the Frechet ideal) via matrix iterations of <θ-ultrafilter-linked posets (restricted to some level of the matrix). This is applied to prove consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cichoń's diagram can be separated into 10 different values.
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DOI 10.1016/j.apal.2020.102856
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References found in this work BETA

Mad Families, Splitting Families and Large Continuum.Jörg Brendle & Vera Fischer - 2011 - Journal of Symbolic Logic 76 (1):198 - 208.
Matrix Iterations and Cichon’s Diagram.Diego Alejandro Mejía - 2013 - Archive for Mathematical Logic 52 (3-4):261-278.

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Citations of this work BETA

Cichoń’s diagram and localisation cardinals.Martin Goldstern & Lukas Daniel Klausner - 2021 - Archive for Mathematical Logic 60 (3):343-411.

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