# 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.