On Monochromatic Solutions of Linear Equations Using At Least Three Colors

TL;DR

This paper investigates the occurrence of monochromatic solutions to linear equations in colored sets, focusing on the case of at least three colors and establishing properties of $r$-commonness and $r$-uncommonness.

Contribution

It introduces the concept of $r$-commonness for linear equations with odd terms and proves that 2-uncommon equations are also $r$-uncommon for all $r extgreater=3.

Findings

Linear equations with an odd number of terms have specific $r$-commonness properties.

Any 2-uncommon linear equation remains $r$-uncommon for all $r extgreater=3.

Results extend understanding of monochromatic solutions in multi-color settings.

Abstract

We study the number of monochromatic solution to linear equation in $\{1,\dots,n\}$ when we color the set by at least three colors. We consider the $r$-commonness for $r\geq 3$ of linear equation with odd number of terms, and we also prove that any $2$-uncommon equation is $r$-uncommon for any $r\geq 3$.