The adjacent Hindman's theorem and the $\mathbb Z$-Ramsey's theorem

We consider the restriction of Ramsey's theorem that arises from considering only translation-invariant colourings of pairs, and show that this has the same strength (both from the viewpoint of Reverse Mathematics and from the viewpoint of Computability Theory) as the { \em Adjacent Hindman's Theorem}, proposed by L. Carlucci (Arch. Math. Log. { \bf 57} (2018), 381--359). We also investigate some higher dimensional versions of both of these statements.

Categories: math.LO math.CO