Filter-linkedness and its effect on preservation of cardinal characteristics

Jörg Brendle, Miguel A. Cardona, Diego A. Mejía

Published 2018-09-13Version 1

We introduce the property "$F$-linked" of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties "$\mu$-$F$-linked" and "$\theta$-$F$-Knaster" for posets in the natural way. We show that $\theta$-$F$-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 $\theta$-$\mathrm{Fr}$-Knaster posets (where $\mathrm{Fr}$ is the Frechet ideal) via matrix iterations of $<\theta$-ultrafilter-linked posets (restricted to some level of the matrix). This is applied to prove new consistency results about Cicho\'n's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated to it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cicho\'n's diagram can be separated into $10$ different values.