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The higher Cichoń diagram

Tom 252 / 2021

Thomas Baumhauer, Martin Goldstern, Saharon Shelah Fundamenta Mathematicae 252 (2021), 241-314 MSC: Primary 03E35; Secondary 03E17, 03E55. DOI: 10.4064/fm666-4-2020 Opublikowany online: 31 August 2020


For a strongly inacessible cardinal $\kappa $, we investigate the relationships between the following ideals:

(1) the ideal of meager sets in the ${ \lt }\kappa $-box product topology,

(2) the ideal of “null” sets in the sense of [She17],

(3) the ideal of nowhere stationary subsets of a (naturally defined) stationary set $S_{\rm pr }^\kappa \subseteq \kappa $.

In particular, we analyze the provable inequalities between the cardinal characteristics for these ideals, and we give consistency results showing that certain inequalities are unprovable.

While some results from the classical case ($\kappa =\omega $) can be easily generalized to our setting, some key results (such as a Fubini property for the ideal of null sets) do not hold; this leads to the surprising inequality cov(null)${}\le {}$non(null). Also, concepts that did not exist in the classical case (in particular, the notion of stationary sets) will turn out to be relevant.

We construct several models to distinguish the various cardinal characteristics; the main tools are iterations with $\mathord \lt \kappa $-support (and a strong “Knaster” version of $\kappa ^+$-c.c.) and one iteration with ${\le }\kappa $-support (and a version of $\kappa $-properness).


  • Thomas BaumhauerInstitute of Discrete Mathematics and Geometry
    TU Wien
    Wiedner Hauptstraße 8–10
    1040 Wien, Austria
  • Martin GoldsternTU Wien
    Institute of Discrete Mathematics and Geometry
    Wiedner Hauptstraße 8–10
    1040 Wien, Austria
  • Saharon ShelahEinstein Institute of Mathematics
    The Hebrew University of Jerusalem
    Edmond J. Safra Campus, Givat Ram
    Jerusalem, 91904, Israel
    Department of Mathematics
    The State University of New Jersey
    Hill Center – Busch Campus, Rutgers
    110 Frelinghuysen Road
    Piscataway, NJ 08854-8019, U.S.A.

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