I will talk about special \(\aleph_n\)-Aronszajn trees and \(\aleph_n\)-Kurepa trees. The main result I want to present is the consistency of the statement that the following holds for every \(0 < n < \omega\): all \(\aleph_n\)-Aronszajn trees are special, there are such, and there exists no \(\aleph_n\)-Kurepa tree.
The proof needs \(\omega\)-many supercompact cardinals. I will discuss the main ideas of the proof.
This is a joint work with Sy-David Friedman.