More than a Woodin cardinal is required to obtain tree property at two adjacent cardinals. We make a first step toward showing that a much weaker assumption is sufficient for the tree property to hold at all even successor cardinals. Specifically, we show that from \(\omega\)-many weak compacts one can obtain the tree property at all of the \(\aleph_{2n}\)'s and from a cardinal which is strong up to a larger weak compact one can in addition have the tree property at \(\aleph_{\omega+2}\) (\(\aleph_\omega\) strong limit).
This work is joint with Sy-D. Friedman.