Rainbow numbers of $[n]$ for ${\sum }_{i=1}^{k-1}{x}_i=x_k$

Abstract

Consider the set $\{1,2,\dots ,n\}=[n]$ and an equation $eq$. The rainbow number of $[n]$ for $eq$, denoted $\mathrm{rb}([n],eq)$, is the smallest number of colors such that for every exact $\mathrm{rb}([n],eq)$-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 $k\ge 5$.

8 pages; added references, renamed the parameter 'rainbow number', notation has been updated to reflect convention, revised the statement and proof of Lemma 3.1, revised the proof of Proposition 3.4, results are unchanged; updated language, updated formatting, results are unchanged