On minimal colorings without monochromatic solutions to a linear equation

TL;DR

This paper classifies minimal colorings of nonzero rational numbers avoiding monochromatic solutions for specific linear equations, advancing understanding of coloring problems in additive combinatorics.

Contribution

It provides a complete classification of minimal colorings for certain families of linear equations over the rationals, revealing new structures and open problems.

Findings

Classified minimal colorings for E(q,3) with q in {3/2,2,3,4}

Classified minimal colorings for E(2,n) with n in {3,4,5,6}

Classified minimal colorings for x_1+x_2+x_3=4x_4

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.