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 (Z⁺)ᵈ where the iᵗʰ set of coordinates satisfies the iᵗʰ given linear Rado system.