Avoiding Monochromatic Solutions to 3-term Equations

TL;DR

This paper investigates how to color integers to minimize monochromatic solutions of 3-term equations, proving none are 'common' and establishing bounds for specific cases, advancing understanding in combinatorial coloring problems.

Contribution

It proves that no 3-term equations are 'common' and provides a lower bound for a class of these equations, advancing the theory of monochromatic solutions in combinatorics.

Findings

No 3-term equations are common.

Established a lower bound for certain 3-term equations.

Progress in bounds for minimizing monochromatic solutions.

Abstract

Given an equation, the integers $[n] = \{1, 2, \dots, n\}$ as inputs, and the colors red and blue, how can we color $[n]$ in order to minimize the number of monochromatic solutions to the equation, and what is the minimum? The answer is only known for a handful of equations, but much progress has been made on improving upper and lower bounds on minima for various equations. A well-studied characteristic an equation, which has its roots in graph Ramsey theory, is to determine if the minimum number of monochromatic solutions can be achieved (asymptotically) by uniformly random colorings. Such equations are called common. We prove that no 3-term equations are common and provide a lower bound for a specific class of 3-term equations.