  We show that the well-partial orderedness of the finite downwards closed
subsets of $\mathbb{N}^k$ ,ordered by inclusion, is equivalent to the
well-foundedness of the ordinal $\omega^{\omega^\omega}$. This was conjectured
to be the case by Hatzikiriakou and Simpson. Since we use Friedman's adjacent
Ramsey theorem for fixed dimensions in the upper bound, we also give a
treatment of the reverse mathematical status of that theorem.
