The adjacent Hindman's theorem and the -Ramsey's theorem
arXiv:2412.14558 · doi:10.1007/s00153-026-01004-8
Abstract
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.
17 pages, only a few minor mistakes from previous version were corrected