On Monochromatic Solutions of Linear Equations Using At Least Three Colors

Laurence P. Wijaya

Published 2025-01-28, updated 2025-02-04Version 2

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 $r\geq 3$.