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

2024/12/19 by Bruno Fernando Aceves-Martínez, Aceves-Martínez, Bruno Fernando, David J. Fernández-Bretón +5
Mathematics · #math.CO #math.LO #msc:03D30 #msc:03F35 #msc:05D10

paper · pdf · doi:10.48550/arxiv.2412.14558

published as Archive for Mathematical Logic 65 (2026), 451-463 · 17 pages, only a few minor mistakes from previous version were corrected

arxiv created 2026/02/24 · arxiv updated 2026/07/30

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.

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