Rainbow Numbers for the Generalized Schur Equation $x_1+x_2+\cdots +x_{m-1}=x_m$

Abstract

We consider the rainbow Schur number $R{S}_m(n)$, defined to be the minimum number of colors such that every coloring of $\{1,2,\dots ,n\}$, using all $R{S}_m(n)$ colors, contains a rainbow solution to the equation $x_1+x_2+\cdots +x_{m-1}=x_m$. Recently, the exact values of $R{S}_3(n)$ and $R{S}_4(n)$ were determined for all $n$. In this paper, we expand upon this work by providing a formula for $R{S}_m(n)$ that holds for all $m\ge 4$ and all $n$. A weakened version of the rainbow Schur number is also considered, for which one seeks solutions to the above-mentioned linear equation where, for a fixed $t\le m$, at least $t$ colors are used.