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arXiv · 2412.14558

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

Bruno Fernando Aceves-Martínez·David J. Fernández-Bretón·L. F. Romero-García·Luis F. Villagómez-Canela
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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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Bruno Fernando Aceves-Martínez, David J. Fernández-Bretón, L. F. Romero-García, Luis F. Villagómez-Canela. 2024-12-19. The adjacent Hindman's theorem and the $\mathbb Z$-Ramsey's theorem. https://doi.org/10.1007/s00153-026-01004-8

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