Elementary proof yielding injective monochromatic solutions

Develop an alternative elementary proof of Rado’s theorem for linear homogeneous equations that also establishes the existence of injective monochromatic solutions, meaning solutions whose coordinates are pairwise distinct.

Background

The proof presented in the paper establishes monochromatic solutions but does not ensure that the coordinates are pairwise distinct. The authors observe that injective monochromatic solutions are usually obtained using van der Waerden’s theorem or by reformulating the problem as a system of linear equations. They explicitly ask whether an alternative elementary proof can provide this stronger conclusion.