COFINAL TYPES BELOW

Abstract

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 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.