The adjacent Hindman's theorem and the $\mathbb Z$-Ramsey's theorem
Logic
2026-02-25 v3 Combinatorics
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.
Keywords
Cite
@article{arxiv.2412.14558,
title = {The adjacent Hindman's theorem and the $\mathbb Z$-Ramsey's theorem},
author = {Bruno Fernando Aceves-Martínez and David J. Fernández-Bretón and L. F. Romero-García and Luis F. Villagómez-Canela},
journal= {arXiv preprint arXiv:2412.14558},
year = {2026}
}
Comments
17 pages, only a few minor mistakes from previous version were corrected