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

Indestructibility of some compactness principles over models of PFA