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arXiv · 2205.00023

Cofinal types below $\aleph_\omega$

Roy Shalev
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Abstract

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

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Roy Shalev. 2022-04-29. Cofinal types below $\aleph_\omega$. https://arxiv.org/abs/2205.00023

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