Almost Kurepa Suslin trees and destructibility of the Guessing Model Property

TL;DR

This paper demonstrates the consistency of the Guessing Model Principle at  with the existence of almost Kurepa Suslin trees and explores the destructibility of this principle via ccc forcing, advancing understanding of set-theoretic combinatorics.

Contribution

It establishes the consistency of the Guessing Model Principle with almost Kurepa Suslin trees and analyzes its destructibility under ccc forcing.

Findings

Guessing Model Principle is consistent with almost Kurepa Suslin trees.

The principle can be destroyed by ccc forcing of size .

Existence of a weak Kurepa tree with failure of the Kurepa Hypothesis.

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 $\omega_2$.