Strongly increasing sequences

Abstract: Using a variation of Woodin's $\mathbb{P}<em>{\mathrm{max}}$ forcing, we force over a model of the Axiom of Determinacy to produce a model of ZFC containing a very strongly increasing sequence of length $\omega</em>{2}$ consisting of functions from $\omega$ to $\omega$. We also show that there can be no such sequence of length $\omega_{4}$.