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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 i^th set of coordinates satisfies the i^th given linear Rado system.
Aaron Robertson (2024) studied this question.