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 $\mathrm{cov}({\mathcal{N}}^s({\mathbb{R}}^d))$ ($0<s<d,d\in \omega$) are equal and all uniformity numbers $\mathrm{non}({\mathcal{N}}^s({\mathbb{R}}^d))$ ($0<s<d,d\in \omega$) are equal. This is a partial answer to Problem 5.3 and 5.4 of [4].

I found the gap in the proof of the main theorem (Theorem 13)