Solution to Hart–van Mill’s Problem 61

A natural question, appeared as Problem 61 in Hart and van Mill's list of open problems on $\beta\omega$ (2024), asks whether every finite partial order is isomorphically embeddable in the Rudin–Keisler order on (types of) ultrafilters over a countable set. Although the positive answer, even for all countable partial orders, was obtained under CH in Blass' thesis (1970), the situation in ZFC alone has remained widely open. The solution was obtained by the speaker jointly with Poliakov (2025). We show that ZFC is sufficient not only to re-prove Blass' result but to prove a much stronger fact: the lattices of finite subsets of a set of cardinality $2^{\mathfrak c}$, and of countable subsets of a set of cardinality $\aleph_1$, both ordered by inclusion, are embeddable in ultrafilters with any relation lying between the Rudin–Keisler and Comfort orders.

References

1. K. P. Hart, J. van Mill, “Problems on $\beta\mathbb{N}$”, Topology Appl., 364:1 (2025), 109092, 24 pp., arXiv: 2205.11204
2. N. L. Poliakov, D. I. Saveliev, On embedding of partially ordered sets in $(\beta\omega,\le_{RK})$, 2025, arXiv: 2511.19354
3. N. L. Poliakov, D. I. Saveliev, “Solution to Hart–van Mill's Problem 61”, UMN, 81:1(487) (2026), 205–206     [N. L. Poliakov, D. I. Saveliev, “Solution to Hart–van Mill's Problem 61”, Russ. Math. Surv., 81:1(487) (2026), 205–206 (to appear)]