In the talk we will focus on the indestructibility of the negation of the weak Kurepa Hypothesis ($\sf \neg wKH$) and the Guessing Model Property ($\sf GMP$) by $\sigma$-centered and Cohen forcings, respectively. Recall that both principles are consequences of $\sf PFA$.
\begin{itemize}
\item We show that over any model of $\sf GMP$, $\sf \neg wKH$ is preserved by any $\sigma$-centered forcing.
\item It follows $\sf \neg wKH$ is preserved over models of $\sf GMP$ by Cohen forcing at $\omega$ of an arbitrary length (because any counterexample would be forced by Cohen forcing of length $\omega_1$ and this is $\sigma$-centered). The result for Cohen forcing can be extended to $\sf GMP$ itself: we show that $\sf GMP$ is preserved by Cohen forcing at $\omega$ of an arbitrary length. Our argument is a generalization of a previous argument of Menachem Magidor who showed it for a single Cohen forcing.
\end{itemize}
In particular this implies that both the tree property at $\omega_2$ and $\sf \neg wKH$ are preserved by Cohen forcing at $\omega$ of an arbitrary length over any model of $\sf PFA$.
This is joint work with Radek Honzik and Chris Lambie-Hanson.