On minimal colorings without monochromatic solutions to a linear equation

Abstract

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.

14 pages, 1 very complicated table. This article originally appeared in the journal INTEGERS in 2007. This version differs from that journal version only in formatting. As such, note that the author contact info is outdated and at least one conjecture contained within has been resolved