Separating Many Localisation Cardinals on the Generalised Baire Space

TL;DR

This paper constructs a large family of functions and their localisation numbers on the generalized Baire space, demonstrating the consistency of assigning different localisation numbers to various functions via forcing.

Contribution

It introduces a method to simultaneously control localisation numbers for multiple functions on the generalized Baire space, answering an open question in the field.

Findings

Constructed a $ ext{k}$-sized family of functions with distinct localisation numbers.

Used a cofinality-preserving forcing to show the consistency of various localisation number assignments.

Resolved an open problem about the variability of localisation numbers for different functions.

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 .