### Essential Kurepa trees versus essential Jech---Kunen trees

by Jin and Shelah. [JiSh:498]

Annals Pure and Applied Logic, 1994

By an omega_1 --tree we mean a tree of size omega_1 and
height omega_1 . An omega_1 --tree is called a Kurepa tree if
all its levels are countable and it has more than omega_1
branches. An omega_1 --tree is called a Jech--Kunen tree if it has
kappa branches for some kappa strictly between omega_1 and
2^{omega_1} . 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 contains no Kurepa subtrees. In
this paper we prove that (1) it is consistent with CH and
2^{omega_1}> omega_2 that there exist essential Kurepa trees and
there are no essential Jech--Kunen trees, (2) it is consistent with
CH and 2^{omega_1}> omega_2 plus the existence of a Kurepa tree
with 2^{omega_1} 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 2^{omega_1}
branches in order to avoid triviality.

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