Extending earlier work of the reviewer, the author presents a necessary and sufficient condition for a system \(Ax=0\) of homogeneous linear equations to have a solution consisting of distinct monochromatic elements in every coloring of \(R\) by \(\kappa\) colors (\(R\) is a subring of the complex field, \(\kappa\) is an infinite cardinal). It is \(| R| \geq \kappa^{+s}\) where \(s\) is the least number of partitions \({\mathcal P}_1,\dots,{\mathcal P}_s\) of the columns of \(A\) such that the sum of the columns in each class of each \({\mathcal P}_i\) is \(0\), and any two columns of \(A\) are separated by some \({\mathcal P}_i\). The proof uses some novel arguments of hypergraph partitions.