Two inequalities between cardinal invariants

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $ω$ of asymptotic density $0$. We obtain an upper bound on the $\ast$-covering number, sometimes also called the weak covering number, of this ideal by proving in Section \ref{sec:covz0} that ${\mathord{\mathrm{cov}}}^{\ast}({\mathcal{Z}}_{0}) \leq \mathfrak{d}$. In Section \ref{sec:skbk} we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove in sharp contrast to the case when $κ= ω$, that if $κ$ is any regular uncountable cardinal, then ${\mathfrak{s}}_κ \leq {\mathfrak{b}}_κ$.

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