Optimization Tools for Computing Colorings of $[1,\cdots ,n]$ with Few Monochromatic Solutions on $3$-variable Linear Equations

TL;DR

This paper introduces optimization methods to compute colorings of integer intervals that minimize monochromatic solutions for certain linear equations, advancing the understanding of Ramsey-type problems.

Contribution

It develops new integer and semidefinite optimization tools to find optimal or near-optimal 2-colorings minimizing monochromatic solutions for 3-variable equations and extends results to more colors.

Findings

Optimized colorings with fewer monochromatic solutions for specific equations.

New bounds and methods for 3-color Schur equations.

Extension of results to more than three colors.

Abstract

A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge R_k(E)$, there exists at least one monochromatic solution. But one can further ask, how many monochromatic solutions is the minimum possible in terms of $n$? Several authors have estimated this function before, here we offer new tools from integer and semidefinite optimization that help find either optimal or near optimal 2-colorings minimizing the number of monochromatic solutions of several families of 3-variable non-regular homogeneous linear equations. In the last part of the paper we further extend to three and more colors for the Schur equation, improving earlier work.