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
TL;DR
The paper investigates two restricted Ramsey-type principles—the $\mathbb Z$-invariant Ramsey's theorem $\mathbb Z\text{-}\mathsf{RT}^n$ and Carlucci's Adjacent Hindman's Theorem $\mathsf{AHT}$—through computability and Reverse Mathematics, using strong and ordinary Weihrauch reducibility. It establishes a unified higher-dimensional picture: for all $d\ge 1$, $\mathbb Z\text{-}\mathsf{RT}^{d+1}$ is strongly Weihrauch-equivalent to $\mathsf{AHT}^d$, with the base case $d=1$ giving $\mathbb Z\text{-}\mathsf{RT}^2 \equiv_{\mathrm{sW}} \mathsf{AHT}$, and extends these equivalences to higher dimensions via explicit reduction functionals. The authors introduce separation and apartness variants, proving $\mathrm{sep}\mathbb Z\text{-}\mathsf{RT}^{d+1} \equiv_{\mathrm{sW}} \mathrm{ap}\mathsf{AHT}^d$ and showing corresponding implications between these restricted Ramsey-type principles and ordinary RT variants, including RM-strength collapses and lower bounds. A key lower bound shows $\mathsf{AHT}^2$ implies $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$, illustrating high computational strength for a seemingly simple combinatorial statement, and the paper highlights open questions about bounded-colour versions and further separations.
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.