A Multidimensional Rado Theorem

TL;DR

This paper extends Rado's theorem to multidimensional positive integer lattices, guaranteeing monochromatic solutions in finite colorings where each coordinate set satisfies a linear Rado system.

Contribution

It generalizes Deuber's theorem to higher dimensions, establishing a multidimensional Rado theorem for positive integer lattices.

Findings

Proves a multidimensional Rado theorem for $ig( Z^+ig)^d$

Guarantees monochromatic solutions in all finite colorings

Extends Deuber's theorem to multidimensional settings

Abstract

We extend Deuber's theorem on $(m,p,c)$-sets to hold over the multidimensional positive integer lattices. This leads to a multidimensional Rado theorem where we are guaranteed monochromatic multidimensional points in all finite colorings of $\left(\mathbb{Z}^+\right)^d$ where the $i^{\mathrm{th}}$ set of coordinates satisfies the $i^{\mathrm{th}}$ given linear Rado system.