Specializing Trees with Small Approximations I

Assuming \(\rm PFA\), we shall use internally club \(\omega_1\)-guessing models as side conditions to show that for every tree \(T\) of height \(\omega_2\) without cofinal branches, there is a proper and \(\aleph_2\)-preserving forcing notion with finite conditions which specialises \(T\). Moreover, the forcing has the \(\omega_1\)-approximation property.