Abstract
Let n be a natural number, 2 <= n < ω. We show that it is consistent to have a model of set theory where aleph_omega is strong limit, 2^{aleph_omega}= aleph_{omega+n}, and the tree property holds at aleph_{omega+2}; we use a hypermeasurable cardinal of an appropriate degree for the result and a variant of the Mitchell forcing followed by the Prikry forcing with collapses.