Published June 30, 2023
| Version v1
Journal article
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Rainbow Numbers of \([m]\times [n]}\) for \(x_1+x_2 = x_3\)
- 1. Iowa State University
- 2. University of Wisconsin-La Crosse
- 3. University of Vermont
Description
Consider the set $[m]\times [n] = \{(i,j)\, : 1\le i \le m, 1\le j \le n\}$ and the equation $eq$: $x_1+x_2 = x_3$.
The \emph{rainbow number} of $[m] \times [n]$ for $eq$, denoted $\rb([m]\times [n],eq)$, is the smallest number of colors such that for every surjective $\operatorname{rb}([m]\times[n], eq)$-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 $\operatorname{rb}(\mn, eq) = m+n+1$ for all values of $m$ and $n$ that are greater than or equal to two.
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