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Two inequalities between cardinal invariants

Volume 237 / 2017

Dilip Raghavan, Saharon Shelah Fundamenta Mathematicae 237 (2017), 187-200 MSC: 03E17, 03E55, 03E05, 03E20. DOI: 10.4064/fm253-7-2016 Published online: 1 December 2016


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 ${\mathfrak {s}}_{\kappa } \leq {\mathfrak {b} }_{\kappa }$.


  • Dilip RaghavanDepartment of Mathematics National University of Singapore
    Singapore 119076
  • Saharon ShelahInstitute of Mathematics
    The Hebrew University
    Jerusalem 9190401, Israel

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