Filter-linkedness and its effect on preservation of cardinal characteristics

Abstract

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 a 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 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.

30 pages, 7 figures