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

References in corpus (11)

• Are a and d your cup of tea?
• Covering of the null ideal may have countable cofinality
• Matrix iterations and Cichon's diagram
• Matrix iterations with vertical support restrictions
• Cichoń's maximum
• Another ordering of the ten cardinal characteristics in Cichoń's diagram
• Template iterations with non-definable ccc forcing notions
• On cardinal characteristics of Yorioka ideals
• The left side of Cichoń's diagram
• Compact Cardinals and Eight Values in Cichoń's Diagram
• Controlling classical cardinal characteristics while collapsing cardinals

Cited by in corpus (9)

• Matrix iterations with vertical support restrictions
• Cichoń's Diagram and Localisation Cardinals
• Set-Theoretic Blockchains
• Many Different Uniformity Numbers of Yorioka Ideals
• Preservation of splitting families and cardinal characteristics of the continuum
• A note on "Another ordering of the ten cardinal characteristics in Cichoń's Diagram" and further remarks
• Uniformity numbers of the null-additive and meager-additive ideals
• Yorioka's characterization of the cofinality of the strong measure zero ideal and its independency from the continuum
• Cichoń's maximum with evasion number