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<p>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 <i>Adjacent Hindman’s Theorem</i>, proposed by L. Carlucci (Arch. Math. Log. <b>57</b> (2018), 381–359). We also investigate some higher dimensional versions of both of these statements.</p>

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The adjacent Hindman’s Theorem and the \(\mathbb Z\)-Ramsey’s Theorem

  • 期刊论文

  • 发表日期：

    2026年3月10日

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  • Bruno Fernando Aceves-Martínez,
  • David J. Fernández-Bretón,
  • L. F. Romero-García,
  • Luis F. Villagómez-Canela

摘要

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 Adjacent Hindman’s Theorem, proposed by L. Carlucci (Arch. Math. Log. 57 (2018), 381–359). We also investigate some higher dimensional versions of both of these statements.