ESSENTIAL KUREPA TREES VERSUS ESSENTIAL JECH-KUNEN TREES

By an omega1-tree we mean a tree of cardinality omega1 and height omeg a1. An omega1-tree is called a Kurepa tree if all its levels are count able and it has more than omega1 branches. An omega1-tree is called a Jech-Kunen tree if it has kappa branches for some kappa Strictly betwe en omega1 and 2omega1. A Kurepa tree is called an essential Kurepa tre e if it contains no Jech-Kunen subtrees. A Jech-Kunen tree is called a n essential Jech-Kunen tree if it contains no Kurepa subtrees. In this paper we prove that (1) it is consistent with CH and 2omega1 > omega2 that there exist essential Kurepa trees and there are no essential Je ch-Kunen trees, (2) it is consistent with CH and 2omega1 > omega2 plus the existence of a Kurepa tree with 2omega1 branches that there exist essential Jech-Kunen trees and there are no essential Kurepa trees. I n the second result we require the existence of a Kurepa tree with 2om ega1 branches in order to avoid triviality.