Rainbow Numbers for the Generalized Schur Equation $x_1 + x_2 + \cdots + x_{m-1} = x_m$
Mark Budden, Bruce Landman
TL;DR
This work determines the rainbow Schur numbers for the generalized Schur equation $E_m: x_1+\cdots+x_{m-1}=x_m$ by proving the exact formula $RS_m(n)=\left\lceil\frac{(m-3)n+\frac{m(m-1)}{2}}{m-2}\right\rceil$ for all $m\ge 4$ and $n\ge\frac{m(m-1)}{2}$, built from a proven lower bound and a parity-aware inductive upper bound. It also introduces weakened rainbow Schur numbers $RS_{t,m}(n)$, establishing a closed form $RS_{t,m}(n)=\left\lceil\frac{(t-3)n+\frac{t(t-1)}{2}+m-t}{t-2}\right\rceil$ for $3\le t\le m$ (with $RS_{2,m}(n)=2$ for large enough $n$), thereby generalizing the rainbow framework to color-usage constraints. Together, these results extend the known $RS_3(n)$ and $RS_4(n)$ to all $m\ge 4$ and connect rainbow and anti-Ramsey perspectives in a unified setting. The findings have implications for anti-Ramsey theory and the study of rainbow solutions in linear systems.
Abstract
We consider the rainbow Schur number $RS_m(n)$, defined to be the minimum number of colors such that every coloring of $\{1,2,\ldots,n\}$, using all $RS_m(n)$ colors, contains a rainbow solution to the equation $x_1+x_2+\cdots +x_{m-1}=x_m$. Recently, the exact values of $RS_3(n)$ and $RS_4(n)$ were determined for all $n$. In this paper, we expand upon this work by providing a formula for $RS_m(n)$ that holds for all $m \geq 4$ 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 $t \leq m$, at least $t$ colors are used.