Two inequalities between cardinal invariants

Dilip Raghavan¹ ; Saharon Shelah²

¹Department of Mathematics National University of Singapore Singapore 119076
²Institute of Mathematics The Hebrew University Jerusalem 9190401, Israel

Fundamenta Mathematicae, Tome 237 (2017) no. 2, pp. 187-200

Résumé

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 that ${\rm cov }^{\ast }({{\mathcal {Z}}}_{0}) \leq {\mathfrak {d}}$. Next, we investigate the relationship between the bounding and splitting numbers at regular uncountable cardinals. We prove that, in sharp contrast to the case when $\kappa = \omega $, if $\kappa $ is any regular uncountable cardinal, then ${\mathfrak {s}}_{\kappa } \leq {\mathfrak {b} }_{\kappa }$.

Détail

DOI : 10.4064/fm253-7-2016

Keywords: prove mathrm zfc inequalities between cardinal invariants first inequality involves cardinal invariants associated analytic p ideal particular ideal subsets omega asymptotic density nbsp obtain upper bound ast covering number sometimes called weak covering number ideal proving cov ast mathcal leq mathfrak investigate relationship between bounding splitting numbers regular uncountable cardinals prove sharp contrast kappa omega kappa regular uncountable cardinal mathfrak kappa leq mathfrak kappa