  For a ring R and system L of linear homogeneous equations, we call a coloring
of the nonzero elements of R minimal for L if there are no monochromatic
solutions to L and the coloring uses as few colors as possible. For a rational
number q and positive integer n, let E(q,n) denote the equation
$\sum_{i=0}^{n-2} q^{i}x_i = q^{n-1}x_{n-1}$. We classify the minimal colorings
of the nonzero rational numbers for each of the equations E(q,3) with q in
{3/2,2,3,4}, for E(2,n) with n in {3,4,5,6}, and for x_1+x_2+x_3=4x_4. These
results lead to several open problems and conjectures on minimal colorings.
