Easton's theorem for the tree property below aleph_omega

Abstract

Starting with infinitely many supercompact cardinals, we show that the tree property at every cardinal $\aleph_n$, $1 < n <\omega$, is consistent with an arbitrary continuum function below $\aleph_\omega$ which satisfies $2^{\aleph_n} > \aleph_{n+1}$, $n<\omega$. Thus the tree property has no provable effect on the continuum function below $\aleph_\omega$ except for the restriction that the tree property at $\kappa^{++}$ implies $2^\kappa>\kappa^+$ for every infinite $\kappa$.