Strongly increasing sequences

Using a variation of Woodin’s ${\mathbb{P}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ 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 ${\omega }_2$ consisting of functions from $\omega$ to $\omega$. We also show that there can be no such sequence of length ${\omega }_4$.