Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

Abstract

Building on recent work of Krueger and the second author, we prove the consistency of the Guessing Model Principle at ${\omega }_2$ together with the existence of an almost Kurepa Suslin tree. In particular, it is consistent that the Guessing Model Principle holds but is destructible by a ccc forcing of size ${\omega }_1$. We also prove the consistency of the existence of a weak Kurepa tree together with the failure of the Kurepa Hypothesis and a certain guessing model principle that, for example, implies the tree property at ${\omega }_2$.