Rainbow Numbers for the Generalized Schur Equation $x_1 + x_2 + \cdots + x_{m-1} = x_m$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917567696732160 |
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| author | Budden, Mark Landman, Bruce |
| author_facet | Budden, Mark Landman, Bruce |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_07357 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rainbow Numbers for the Generalized Schur Equation $x_1 + x_2 + \cdots + x_{m-1} = x_m$ Budden, Mark Landman, Bruce Combinatorics Number Theory 05D10, 11B75 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. |
| title | Rainbow Numbers for the Generalized Schur Equation $x_1 + x_2 + \cdots + x_{m-1} = x_m$ |
| topic | Combinatorics Number Theory 05D10, 11B75 |
| url | https://arxiv.org/abs/2401.07357 |