We prove that the Tree Property for \(\omega_2\) together with \(\mathrm(\omega_1)\) is equiconsistent with the existence of a weakly compact cardinal. Also, we prove that the tree property for \(\omega_2\) together with \(\mathrm\) is equiconsistent with the existence of a weakly compact cardinal which is also reflecting. Similarly, we show that the Special Tree Property for \(\omega_2\) together with \(\mathrm(\omega_1)\) is equiconsistent with the existence of a Mahlo cardinal and the special tree property for \(\omega_2\) together with \(\mathrm\) is equiconsistent with the existence of a Mahlo cardinal which is also reflecting.
This is a joint work with Sy Friedman.