On minimal colorings without monochromatic solutions to a linear equation

Details On minimal colorings without monochromatic solutions to a linear equation On minimal colorings without monochromatic solutions to a linear equation

2010-09-21

arxiv.org/abs/1009.4234v1

Integers, the Electronic Journal of Combinatorial Number Theory 7 (2007), no. 2, 20 pp. Also in a collection: Combinatorial Nu

Mathematics

Combinatorics

14 pages, 1 very complicated table. This article originally appeared in the journal INTEGERS in 2007. This version differs fro

Scientific paper

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.

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