In this talk we aim to compare Kurepa trees and Aronszajn trees. Moreover, we talk about the affect of large cardinal assumptions on this comparison. Using the the method of walks on ordinals, we will show it is consistent with ZFC that there is a Kurepa tree and every Kurepa tree contains a Souslin subtree, if there is an inaccessible cardinal. This is stronger than Komjath's theorem which asserts the same consistency from two inaccessible cardinals. We will briefly sketch the ideas to prove that our large cardinal assumption is optimal. If time permits, we talk about the comparison of Kurepa trees and Aronszajn trees in the presence of no large cardinal.
This is a joint work with Stevo Todorcevi