Abstract
Starting with modest large cardinal assumptions (a hypermeasurable cardinal of a sufficient degree) we construct a model where $\aleph_\omega$ is a strong limit cardinal, the tree property holds at $\aleph_{\omega+2}$, and $2^{\aleph_\omega} = \aleph_{\omega+n}$ for any fixed $2 \le n < \omega$. The proof uses a variant of the Mitchell forcing and is based on a product-style analysis due to Abraham.
The results are joint with Sy D.\ Friedman and R.\ Honzik.