On monochromatic solutions to linear equations over the integers

TL;DR

This paper investigates the frequency of monochromatic solutions to linear equations in two-colorings of integers, establishing that nontrivial equations have a constant fraction of solutions and analyzing the commonness of specific four-term equations.

Contribution

It proves that all nontrivial linear equations have a constant fraction of monochromatic solutions in any 2-coloring of integers and disproves a conjecture about the uncommonness of a specific four-term equation.

Findings

Any nontrivial linear equation has a constant fraction of monochromatic solutions.

The four-term equation x_1 + 2x_2 - x_3 - 2x_4 = 0 is uncommon over 1,..., n.

Disproved a conjecture by Costello and Elvin regarding the commonness of certain equations.

Abstract

We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of $\{1,\ldots,n\}$. We further study commonness of four-term equations and disprove a conjecture of Costello and Elvin by showing that, unlike over $\mathbb{F}_p$, the four-term equation $x_1 + 2x_2 - x_3 - 2x_4 = 0$ is uncommon over $\{1,\ldots,n\}$.