2022/04/29 by Roy Shalev, Shalev, Roy
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO) #Mathematical Dynamics and Fractals #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2205.00023
openalex publication_date 2022/04/29 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
It is proved that for every positive integer n, the number of non-Tukey-equivalent directed sets of cardinality ≤ ℵn is at least cn+2, the (n+2)-Catalan number. Moreover, the Tukey class \mathcal Dℵn of directed sets of cardinality ≤ ℵn 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.