We continue our study of the localization cardinals in generalized Baire space from last week. The main goal of this week is to show consistency results complimenting the ZFC results from last week. Specifically, we aim to show that the ZFC bounds found the previous week are sharp for measurable cardinals. For instance we will show, relative to a supercompact cardinal \(\kappa\) that it is consistent that \(\kappa\) is supercompact (in particular measurable) and \(\mathfrak{b}_{Id^+}(\in^*) = \kappa^{++} = 2^\kappa\) holds. Here \(Id^+\) denotes the function mapping \(\alpha \mapsto \alpha^+\). More generally we will show the consistency of \(\mathfrak{b}_{Id^+\xi}(\in^*) = \kappa^{+\xi+1} = 2^\kappa\) for any \(\xi\) as well as dual results for \(\mathfrak{d}_{Id^{+\xi}}(\in^*)\) for supercompact cardinals. These arguments answer questions of Brendle, Brooke-Taylor, Friedman, Montoya and van der Vlugt. Notably we show the consistency (modulo a supercompact) of \(\mathfrak{b}_\kappa (\in^*)< \mathfrak{b}_{Id^+}(\in^*)\). Time permitting, we will also show that the club-capturing versions of these cardinals can be forced to be distinct at inaccessibles, and, moreover that this forcing will preserve supercompactness should it hold. If even more time permits we will discuss more consistent inequalities to other cardinal characteristics on a measurable such as \(\mathfrak{p}_\kappa\) and \(\mathfrak{s}_\kappa\).

This is a series of two talks. In the last week talk we have discussed the ZFC results and sketch the general landscape. This week we will prove consistency results complementing these.

This is (still) joint work with Tom Benhamou.