Essential Kurepa trees versus essential Jech-Kunen trees (Q1337358)

Am \(\omega_ 1\)-tree is called a Kurepa tree if all its levels are countable and it has more than \(\omega_ 1\) branches; it is called a Jech-Kunen tree if it has \(\kappa\) branches for some \(\kappa\) with \(\omega_ 1 < \kappa < 2^{\omega_ 1}\). A Kurepa tree is said to be essential if it contains no Jech-Kunen subtree, and a Jech-Kunen tree is said to be essential if it contains no Kurepa subtree. In this paper the authors show that it is consistent with \(\text{CH }+2^{\omega_ 1} > \omega_ 2\) that there exist essential Kurepa trees but no essential Jech-Kunen trees, and it is consistent with \(\text{CH }+2^{\omega_ 1} > \omega_ 2\) + (there is a Kurepa tree with \(2^{\omega_ 1}\) branches) that there exist essential Jech-Kunen trees but no essential Kurepa trees.