# Rainbow numbers of $[n]$ for $\sum_{i=1}^{k-1} x_i = x_k$

## TL;DR

This paper determines the minimum number of colors needed to guarantee a rainbow solution to specific linear equations within the set [n], focusing on the equations  = x_k for small k and providing bounds for larger k.

## Contribution

It establishes exact rainbow numbers for the equations  = x_k for k=3,4, and provides a general lower bound for larger k.

## Key findings

- Rainbow number for  = x_k with k=3,4 determined.
- A general lower bound for rainbow numbers when k  established.
- Results contribute to understanding rainbow solutions in combinatorics.

## Abstract

Consider the set $\{1,2,\dots,n\} = [n]$ and an equation $eq$. The rainbow number of $[n]$ for $eq$, denoted $\operatorname{rb}([n],eq)$, is the smallest number of colors such that for every exact $\operatorname{rb}([n], eq)$-coloring of $[n]$, there exists a solution to $eq$ with every member of the solution set assigned a distinct color. This paper focuses on linear equations and, in particular, establishes the rainbow number for the equations $\sum_{i=1}^{k-1} x_i = x_k$ for $k=3$ and $k=4$. The paper also establishes a general lower bound for $k \ge 5$.