Cofinal types below ${\aleph }_{\omega }$

Abstract

It is proved that for every positive integer $n$, the number of non-Tukey-equivalent directed sets of cardinality $\le {\aleph }_n$ is at least $c_{n+2}$, the $(n+2)$-Catalan number. Moreover, the Tukey class ${\mathcal{D}}_{{\aleph }_n}$ of directed sets of cardinality $\le {\aleph }_n$ 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.

arXiv comment25 pages, 3 figures, update after referee comments. We thank the referee for their effort and for writing a detailed thoughtful report that improved this paper