Cichoń’s diagram for uncountable cardinals

Abstract

We develop a version of Cichoń's diagram for cardinal invariants on the generalized Cantor space 2 κ or the generalized Baire space κ κ where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example we obtain a natural generalization of the Bartoszyński-Raisonnier-Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.

Questions on generalised Baire spaces

“…But the following remains open: Further background. Another related question is the possibility of cov(M κ ) < in(κ) for limit cardinals κ, which was recently proved to be consistent in an argument to appear in [11].…”

“…Theorem 2.14 (Brendle, Brooke-Taylor, Friedman, Montoya; [11]) If κ is strongly inaccessible then all the implications in Figure 2 hold.…”

“…Nevertheless, for strongly inaccessible κ, one can consider generalisations of certain combinatorial characteristics which are equivalent to and in the classical setting. Definition (Brendle, Brooke‐Taylor, Friedman, Montoya; ) Let κ be strongly inaccessible. …”

“…In , several models are produced to separate cardinal invariants in this diagram (e.g., κ‐Cohen forcing increases but leaves small) although many questions remain; cf. Question .…”

Cardinal characteristics at κ in a small u(κ) model

“…If in addition κ is strongly inaccessible we have a similar diagram as in the countable case (For specific details about these properties see [3]): …”