COFINAL TYPES BELOW ?

It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality <= aleph(n) is at least cn+2, the (n + 2)-Catalan number. Moreover, the class D aleph(n) of directed sets of cardinality <=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.