An infinite color analogue of Radoʹs 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 2 ℵ 0 > ℵ s if and only if the equation x 0 + s x 1 = x 2 + ⋯ + x s + 2 is ℵ 0 -regular over R . This generalizes the case s = 1 due to Erdős.