Separating Many Localisation Cardinals on the Generalised Baire Space

Abstract

Given a cofinal cardinal function $h\in{}^\kappa\kappa$ for $\kappa$ inaccessible, we consider the dominating $h$-localisation number, that is, the least cardinality of a dominating set of $h$-slaloms such that every $\kappa$-real is localised by a slalom in the dominating set. It was proved in arXiv:1611.08140 that the dominating localisation numbers can be consistently different for two functions $h$ (the identity function and the power function). We will construct a $\kappa$-sized family of functions $h$ and their corresponding localisation numbers, and use a ${\leq}\kappa$-supported product of a cofinality-preserving forcing to prove that any simultaneous assignment of these localisation numbers to cardinals above $\kappa$ is consistent. This answers an open question from arXiv:1611.08140 .