Filter-linkedness and its effect on preservation of cardinal characteristics

TL;DR

This paper introduces new linkedness properties of posets related to filters, demonstrating their preservation of certain cardinal characteristics and applying these results to consistency proofs in set theory, including Cichoń's diagram.

Contribution

It defines the properties "$F$-linked", "$ heta$-$F$-Knaster", and develops a technique for constructing such posets, leading to new consistency results without large cardinals.

Findings

Preservation of unbounded and almost disjoint families by $ heta$-$F$-Knaster posets.

Construction of $ heta$-$ ext{Fr}$-Knaster posets via matrix iterations.

Consistency of separating Cichoń's diagram into 10 distinct values with three strongly compact cardinals.

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 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 with 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.