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PDF logo On minimal colorings without monochromatic solutions to a linear equation

by Boris Alexeev, Jacob Fox, and Ron Graham

Abstract

For a ring \(R\) and system \(\mathcal L\) of linear homogeneous equations, we call a coloring of the nonzero elements of \(R\) minimal for \(\mathcal L\) if there are no monochromatic solutions to \(\mathcal L\) 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 \(\{\frac 32,2,3,4\}\), 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.

Approximate citation

Boris Alexeev, Jacob Fox, and Ron Graham
On minimal colorings without monochromatic solutions to a linear equation
Combinatorial Number Theory, de Gruyter, Berlin, 2007, pp. 1–22.
Integers the Electronic Journal of Combinatorial Number Theory 7 (2007), no. 2, 20pp.

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