We say that a regular cardinal \(\kappa \ge \omega\) has the tree property if there are no \(\kappa\)‑Aronszajn trees. It is known that if \(2^\omega = \omega_1\), there are \(\omega_2\)‑Aronszajn trees; thus the tree property at \(\omega_2\) implies the negation of CH (and analogously for larger cardinals). All the usual forcings for the tree property at \(\omega_2\), such as the Mitchell forcing or the Sacks forcing, give \(2^\omega = \omega_2\). We show that the “gap two” is no consequence of the tree property: indeed, we show that – starting with infinitely many weakly compact cardinals – the tree property can hold at every even cardinal below \(\aleph_\omega\) and the continuum function below \(\aleph_\omega\) can be arbitrary (such that \(2^{\omega_2n} \ge \omega_{2n+2}, n \lt \omega)\). We prove a similar result for the weak tree property as well (\(\kappa\) has the weak tree property if there are no special \(\kappa\)‑Aronszajn trees).

This work is joint with S. Stejskalova.