Cofinal Types below ℵω

R. O.Y. Shalev

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality <![CDATA[ $\leq \aleph _n$ ]]> is at least <![CDATA[ $c_{n+2}$ ]]>, the <![CDATA[ $(n+2)$ ]]> -Catalan number. Moreover, the class <![CDATA[ $\mathcal D_{\aleph _n}$ ]]> of directed sets of cardinality <![CDATA[ $\leq \aleph _n$ ]]> contains an isomorphic copy of the poset of Dyck <![CDATA[ $(n+2)$ ]]> -paths. Furthermore, we give a complete description whether two successive elements in the copy contain another directed set in between or not.

Original languageEnglish
JournalJournal of Symbolic Logic
StateAccepted/In press - 2023

Bibliographical note

Publisher Copyright:
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic.


The author is supported by the European Research Council (grant agreement ERC-2018-StG 802756).

FundersFunder number
European Research CouncilERC-2018-StG 802756


    • 03E04
    • 2020 Mathematics Subject Classification


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