Reverse mathematics of the finite downwards closed subsets of ${\mathbb{N}}^k$ ordered by inclusion and adjacent Ramsey for fixed dimension

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.