A Multidimensional Rado Theorem

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 $i^{\mathrm{t}\mathrm{h}}$ set of coordinates satisfies the $i^{\mathrm{t}\mathrm{h}}$ given linear Rado system.