Rainbow numbers of $[m] \times [n]$ for $x_1 + x_2 = x_3$

TL;DR

This paper determines the exact rainbow number for the equation x1 + x2 = x3 over the grid [m]×[n], establishing it as m + n + 1 for all m, n ≥ 2, which guarantees a rainbow solution under any coloring.

Contribution

It provides a precise formula for the rainbow number of [m]×[n] for the equation x1 + x2 = x3, extending understanding of rainbow solutions in combinatorial grid settings.

Findings

Rainbow number equals m + n + 1 for all m, n ≥ 2

Guarantees existence of rainbow solutions in any coloring with that many colors

Extends previous results on rainbow solutions for additive equations

Abstract

Consider the set $[m]\times [n] = \{(i,j)\, : 1\le i \le m, 1\le j \le n\}$ and the equation $x_1+x_2 = x_3$, namely $eq$. The \emph{rainbow number of $[m] \times [n]$ for $eq$}, denoted $\text{rb}([m]\times [n],eq)$, is the smallest number of colors such that for every surjective $\text{rb}([m]\times[n], eq)$-coloring of $[m]\times [n]$ there must exist a solution to $eq$, with component-wise addition, where every element of the solution set is assigned a distinct color. This paper determines that $\text{rb}([m]\times [n], eq) = m+n+1$ for all values of $m$ and $n$ that a greater than or equal to $2$.