In this talk we will consider some recent results regarding the higher Baire spaces analogues of the almost disjointness, bounding, dominating and splitting numbers. In particular, we will discuss the recently established relative consistency of \(\mathfrak{s}(\kappa)=\kappa^+<\mathfrak{b}(\kappa)\) for \(\kappa\)-strongly unfoldable, the relative consistency of \(\kappa^+<\mathfrak{a}(\kappa)=\mathfrak{b}(\kappa)<\mathfrak{d}(\kappa)<\mathfrak{c}(\kappa)\) for \(\kappa\) regular uncountable, as well as the global evaluation of the spectrum of \(\kappa\)-mad families.

The results build to a great extent on the method of non-linear iterations of Cumming-Shelah, the method of matrix iterations as originally appearing in the work on the ultrafilter and dominating numbers of Blass-Shelah and its subsequent developments, and give (among others) an interesting application of Kunen's isomorphism of names argument in the context of Easton forcing. We will conclude the talk with a brief discussion of remaining open questions.