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

TL;DR

This paper demonstrates that a translation-invariant restriction of Ramsey's theorem is equivalent in strength to the Adjacent Hindman's Theorem, exploring their relationship and higher-dimensional variants within Reverse Mathematics and Computability Theory.

Contribution

It establishes the equivalence between a specific form of Ramsey's theorem and the Adjacent Hindman's Theorem, and investigates their higher-dimensional extensions.

Findings

Equivalent strength of translation-invariant Ramsey and Adjacent Hindman's theorems

Analysis within Reverse Mathematics and Computability Theory

Exploration of higher-dimensional versions

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.