Paper detail

Indestructibility of some compactness principles over models of PFA

We show that $\mathsf{PFA}$ (Proper Forcing Axiom) implies that adding any number of Cohen subsets of $ω$ will not add an $ω_2$-Aronszajn tree or a weak $ω_1$-Kurepa tree, and moreover no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree (a tree of height and size $ω_1$ with at least $ω_2$ cofinal branches). This partially answers an open problem whether ccc forcings can add $ω_2$-Aronszajn or $ω_1$-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 $ω$. And moreover, $\mathsf{GMP}$ implies that no $σ$-centered forcing can add a weak $ω_1$-Kurepa tree. For more generality, we study the principle $\mathsf{GMP}$ at an arbitrary regular cardinal $κ= κ^{<κ}$ (we denote this principle $\mathsf{GMP}_{κ^{++}}$), and as an application we show that there is a model in which there are no weak $\aleph_{ω+1}$-Kurepa trees and no $\aleph_{ω+2}$-Aronszajn trees.