By an ω1-tree we mean a tree of cardinality ω1 and height ω1. An ω1-tree is called a Kurepa tree if all its levels are countable and it has more than ω1 branches. An ω1-tree is called a Jech–Kunen tree if it has κ branches for some κ strictly between ω1 and 2ω1. A Kurepa tree is called an essential Kurepa tree if it contains no Jech–Kunen subtrees. A Jech–Kunen tree is called an essential Jech–Kunen tree if it is no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2ω1 #62; ω2 that there exist essential Kurepa trees and there are no essential Jech–Kunen trees, (2) it is consistent with CH and 2ω1 #62; ω2 plus the existence of a Kurepa tree with 2ω1 branches that there exist essential Jech–Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2ω1 branches in order to avoid triviality.
The first author would like to thank Mathematics Department of Rutgers University for its hospitality during his one week visit there in October 1992, when the main part of the paper was developed.
