tree properties on $\omega_1$ and $\omega_2$

I wrote a short note with the consistency proof, which can be found at http://www.math.cmu.edu/users/jcumming/papers/kurepa/kurepa.pdf. It is pretty rough, please tell me if there are problems.

The answer to your question is yes. During the "IPM conference on set theory and model theory" James cummings gave me the basic idea of the proof of the following theorem:

Theorem. Assuming the existence of a weakly compact cardinal, it is consistent that there exists a Kurepa tree and tree property at $\aleph_2$ holds.