If \(\kappa \geq \omega\) is a regular cardinal and \(h:\kappa \to \kappa\) is a function we define an \(h\)-slalom to be a function \(\phi:\kappa \to [\kappa]^{<\kappa}\) so that for all \(\alpha < \kappa\) we have \(|\phi(\alpha)|\leq h(\alpha)\). A function \(f\in \kappa^\kappa\) is eventually captured by such a \(\phi\), in symbols \(f \in^* \phi\) if for all but boundedly many \(\alpha\) we have \(f(\alpha) \in \phi(\alpha)\). The associated cardinal characteristics \(\mathfrak{b}_h(\in^*)\) and \(\mathfrak{d}_h(\in^*)\) were first introduced by Bartoszynski in the classical cass \(\kappa = \omega\). When \(h\) is the identity we use the subscript \(\kappa\). Bartoszynski showed that they are the same for any \(h\) and correspond to the additivity and the cofinality of the Lebesgue measure zero ideal respectively. In their paper on the higher Cichon diagram, Brendle, Brooke-Taylor, Friedman, and Montoya looked at the natural higher analogues. They showed that roughly the same relations and forcing results hold when \(\kappa\) is inaccessible as in the countable case except for one notable exception - \(\mathfrak{d}_{h}(\in^*)\) and \(\mathfrak{d}_{h'}(\in^*)\) can be different for choices of \(h, h'\). This was improved later by van der Vlugt to have \(\kappa^+\) many different values for different \(h\)'s. Left open is whether the same is true for the cardinals \(\mathfrak{b}_h(\in^*)\).

In this talk we will look at the case where \(\kappa\) is measurable. Surprisingly, we show here the situation looks very different. For instance, provably in ZFC, if \(\kappa\) is measurable then \(\mathfrak{b}_\kappa(\in^*) = \kappa^+\) and \(\mathfrak{d}_\kappa(\in^*) = 2^\kappa\). More generally bounds on the values of \(\mathfrak{b}_h(\in^*) = \kappa^+\) and \(\mathfrak{d}_h(\in^*) = 2^\kappa\) can be computed by looking at the \(V\)-cardinality of the ordinal represented by \([h]_U\) in the ultrapower by \(U\) (for a measure \(U\) on \(\kappa\)). This remains true even when we modify the ideal on which we allow errors in capturing.

In this first talk we will discuss these ZFC results and sketch the general landscape. The following week will prove consistency results complementing these.

This is joint work with Tom Benhamou.