An infinite color analogue of Rado's theorem

Let R be a subring of the complex numbers and a be a cardinal. A system L of linear homogeneous equations with coefficients in R is called a-regular over R if, for every a-coloring of the nonzero elements of R, there is a monochromatic solution to L in distinct variables. In 1943, Rado classified those finite systems of linear homogeneous equations that are a-regular over R for all positive integers a. For every infinite cardinal a, we classify those finite systems of linear homogeneous equations that are a-regular over R. As a corollary, for every positive integer s, we have ℵ02>ℵs if and only if the equation x0+sx1=x2+⋯+xs+2 is ℵ0-regular over R. This generalizes the case s=1 due to Erdős.