Recall that the tree property at a regular cardinal \(\kappa\) says that every \(\kappa\)-tree has a cofinal branch, and the weak Kurepa hypothesis at \(\kappa\) says that there exists a tree of size and height \(\kappa\) which has at least \(\kappa^+\) cofinal branches. We will prove that over any transitive model of PFA, the tree property at \(\omega_2\) cannot be destroyed by the single Cohen forcing \(\rm{Add} (\omega,1)\) and the negation of the weak Kurepa hypothesis at \(\omega_1\) cannot be destroyed by a \(\sigma\)-centered forcing.

We will observe that a model-theoretic principle, Guessing model property (GMP), is enough for the preservation results. GMP can be formulated also for larger cardinals. We will give an application of our result by showing that there is a model in which the negation of the weak Kurepa hypothesis holds at \(\aleph_{\omega+1}\).

This talk will be given in mixed mode, in person as well as via Zoom.

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