Two inequalities between cardinal invariants

TL;DR

This paper establishes two ZFC inequalities relating cardinal invariants, including bounds involving analytic P-ideals and relationships between splitting and bounding numbers at uncountable regular cardinals.

Contribution

It proves new inequalities between cardinal invariants for analytic P-ideals and uncountable regular cardinals, expanding understanding of their relationships.

Findings

Upper bound on the weak covering number of the density zero ideal by .

Proved _{} \u2264 _{} for regular uncountable cardinals.

Established inequalities between splitting and bounding numbers at uncountable cardinals.

Abstract

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 $\omega$ 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 $\kappa = \omega$, that if $\kappa$ is any regular uncountable cardinal, then ${\mathfrak{s}}_{\kappa} \leq {\mathfrak{b}}_{\kappa}$.