Indestructibility of some compactness principles over models of PFA

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.