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Abstract  We show that the well partial orderedness of the finite downwards closed subsets of    N  k   , ordered by inclusion, is equivalent to the well foundedness of the ordinal    ω   ω  ω    . 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.

Reverse mathematics of the finite downwards closed subsets of N k ordered by inclusion and adjacent Ramsey for fixed dimension