A model in which every Kurepa tree is thick.
Renling Jin · Notre Dame Journal of Formal Logic · 1991
In this paper we show that, assuming the existence of two strongly inaccessible cardinals, it is consistent with CH (or -ιC7/) plus 2 ωi > ω 2 that there exists a Kurepa tree with 2 ωi -many branches and no ω x -trees have λmany branches for some λ strictly between α>i and 2 ωi .A tree is a partially ordered set (T,) is a subtree of (T 9 ωi.A Kurepa tree Γis called thick if |(B(Γ)| = 2 ωi .An ω^tree is called a Jech-Kunen tree if ωj ω 2 , (1) a Jech-Kunen tree T is a Kurepa tree if |T a \ ω 2 , in which there is a Jech-Kunen tree.In fact, it is a Kurepa tree with less than 2 ωimany branches.The independence of the existence of a Jech-Kunen tree (in terms of a compact Hausdorff space) under CH plus 2 ωι > ω 2 was given by Kunen [6].The detailed proof can be found in Juhasz [5], Theorem 4.8.In Kunen's model all Kurepa trees, including those with 2 ωi -many branches, are also killed.Is it necessary to kill all Kurepa trees when we kill all Jech-Kunen trees?In Jin [4], Kunen proved that it is consistent with CH plus 2 ωi > ω 2 that there is a thick Kurepa tree which has no Jech-Kunen subtrees.So it is natural to ask