Indestructibility of some compactness principles over models of PFA

TL;DR

This paper demonstrates that under PFA, certain compactness principles like the guessing model principle are indestructible by Cohen forcing, preventing the addition of specific trees such as Aronszajn and Kurepa trees, and explores these principles at larger cardinals.

Contribution

It proves the indestructibility of the guessing model principle under Cohen forcing and extends the analysis to arbitrary regular cardinals, addressing open problems about tree existence.

Findings

PFA implies Cohen forcing cannot add ω₂-Aronszajn or weak ω₁-Kurepa trees.

GMP and ISP are preserved under Cohen forcing.

No σ-centered forcing can add weak ω₁-Kurepa trees.

Abstract

We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $\omega$ will not add an $\omega_2$-Aronszajn tree or a weak $\omega_1$-Kurepa tree, and moreover no $\sigma$-centered forcing can add a weak $\omega_1$-Kurepa tree (a tree of height and size $\omega_1$ with at least $\omega_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $\omega_2$-Aronszajn or $\omega_1$-Kurepa trees. We actually prove more: We show that a consequence of $\mathsf{PFA}$, namely the guessing model principle, $\mathsf{GMP}$, which is equivalent to the ineffable slender tree property, $\mathsf{ISP}$, is preserved by adding any number of Cohen subsets of $\omega$. And moreover, $\mathsf{GMP}$ implies that no $\sigma$-centered forcing can add a weak $\omega_1$-Kurepa tree. For more generality, we study the principle $\mathsf{GMP}$ at an arbitrary regular cardinal $\kappa = \kappa^{<\kappa}$ (we denote this principle $\mathsf{GMP}_{\kappa^{++}}$), and as an application we show that there is a model in which there are no weak $\aleph_{\omega+1}$-Kurepa trees and no $\aleph_{\omega+2}$-Aronszajn trees.