Separating covering numbers and separating uniformity numbers of Hausdorff measures need large continuum

Abstract

We show that if $\mathfrak{c} = \aleph_2$ then all covering numbers of Hausdorff measures $\operatorname{cov}(\mathcal{N}^s(\mathbb{R}^d))$ ($0 < s < d, d \in \omega$) are equal and all uniformity numbers $\operatorname{non}(\mathcal{N}^s(\mathbb{R}^d))$ ($0 < s < d, d \in \omega$) are equal. This is a partial answer to Problem 5.3 and 5.4 of [4].