Simultaneous strengthened lemma for systems of equations

Determine whether a strengthened version of Lemma 3.4 can handle multiple equations and different added coefficients simultaneously, thereby supporting an elementary extension of the proof to systems of linear Diophantine equations.

Background

The paper proves the sufficiency of Rado’s condition for a single linear homogeneous Diophantine equation using an inductive lemma that preserves partition regularity when a rational multiple of one variable is added to another. The authors note that the corresponding general theorem concerns finite systems of Diophantine equations, but extending their method would require this lemma to operate simultaneously on multiple equations with different coefficients. They explicitly state that it is unknown whether the required strengthened property holds.