Elementary proof of Rado’s theorem for systems

Establish an elementary proof of the general Rado theorem for systems of linear equations without using van der Waerden’s theorem.

Background

The paper gives an elementary proof, apart from compactness, of the sufficiency of Rado’s condition for a single linear homogeneous equation while avoiding van der Waerden’s theorem. The general formulation of Rado’s theorem characterizes partition regularity for finite systems of Diophantine equations, but the paper does not establish an analogous van der Waerden-free proof for systems. The authors therefore pose this as an explicit question.