I think that in all classical models of TP($\omega_2$) we have $2^{\omega_0}=\omega_2$. Is there a known model of TP($\omega_2$) + $2^{\omega_0}>\omega_2$ at all?
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1$\begingroup$ What is $TP(\omega_2)$? $\endgroup$– Andreas BlassMar 14, 2012 at 18:36
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1$\begingroup$ It would seem to be the tree property: every $\omega_2$-tree has an $\omega_2$-branch. $\endgroup$– Joel David HamkinsMar 14, 2012 at 22:15
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$\begingroup$ Yes, I meant the tree property, sorry! $\endgroup$– Ajdin HalilovicMar 15, 2012 at 11:36
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I think Spencer Unger in "Fragility and indestructibility of the tree property" proves that if one adds an arbitrary number of Cohen reals to Mitchell's model, then the tree property survives (see http://www.math.cmu.edu/~sunger/).