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

Are the following mutually consistent (relative to large cardinals)?

(1) There are no $\omega_2$-Aronszajn trees.

(2) There is an $\omega_1$-Kurepa tree.

In the models I know of the tree property at $\omega_2$, it also holds that there are no weak Kurepa trees on $\omega_1$ (also called Canadian trees).

• What's a weak Kurepa tree? – Paul McKenney Sep 10 '15 at 17:27
• It's a tree of height $\omega_1$ and levels of size at most $\omega_1$ with at least $\omega_2$ branches. Obviously they exist under CH. – Monroe Eskew Sep 10 '15 at 17:43
• Canadian trees? – Asaf Karagila Sep 10 '15 at 19:52
• This says it is a pun too awful to mention. – Miha Habič Sep 10 '15 at 21:08
• Incorrect. There is an $\omega_2$-Aronszajn tree in $L$. – Monroe Eskew Oct 16 '15 at 23:21

## 2 Answers

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.

• Though the proof is not really difficult, but I think it is better we wait for a published version of Cummings work. – Mohammad Golshani Oct 18 '15 at 12:12
• Mohammad, thanks for this information. Given that these publications usually take a long time to come out, could you post a rough sketch of the idea? – Monroe Eskew Oct 18 '15 at 13:30
• Any update? Does this use a new method for the tree property at $\omega_2$ or is the key a new method for forcing Kurepa trees? – Monroe Eskew Jan 16 '16 at 23:46
• @MonroeEskew I have no idea if there is preprint available, but the fact is that the proof if not really difficult. It is simply the product of Mitchell forcing and (a modified version of the) forcing to add a Kurepa tree with $\kappa$-many branches, where $\kappa$ is the weakly compact we start at the beginning. – Mohammad Golshani Jan 17 '16 at 5:48
• It seems like the hard part would be showing that the tree property on $\aleph_2$ is preserved. – Monroe Eskew Jan 17 '16 at 18:10