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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2511.20528 |
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| _version_ | 1866911286537748480 |
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| author | Vestal, Don Sax, Jonathan |
| author_facet | Vestal, Don Sax, Jonathan |
| contents | In 2001, Robertson and Schaal found the 2-color off-diagonal generalized Schur numbers: for two positive integers $k$ and $l$, they determined the smallest positive integer $S = S(k, l)$ such that for any coloring of the integers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1 + x_2 + \dots + x_k = x_0$ or a blue solution to the equation $x_1 + x_2 + \dots + x_l = x_0$. We extend this result to find the continuous version: for two positive integers $k$ and $l$, we find the smallest real number $S = S_\mathbb{R} (k, l)$ such that for any coloring of the real numbers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1 + x_2 + \dots + x_k = x_0$ or a blue solution to the equation $x_1 + x_2 + \dots + x_l = x_0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Off-Diagonal Continuous Rado Numbers $x_1 + x_2 + \dots + x_k = x_0$ Vestal, Don Sax, Jonathan Combinatorics 05D10 In 2001, Robertson and Schaal found the 2-color off-diagonal generalized Schur numbers: for two positive integers $k$ and $l$, they determined the smallest positive integer $S = S(k, l)$ such that for any coloring of the integers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1 + x_2 + \dots + x_k = x_0$ or a blue solution to the equation $x_1 + x_2 + \dots + x_l = x_0$. We extend this result to find the continuous version: for two positive integers $k$ and $l$, we find the smallest real number $S = S_\mathbb{R} (k, l)$ such that for any coloring of the real numbers from 1 to $S$ using red and blue, there must be a red solution to the equation $x_1 + x_2 + \dots + x_k = x_0$ or a blue solution to the equation $x_1 + x_2 + \dots + x_l = x_0$. |
| title | Off-Diagonal Continuous Rado Numbers $x_1 + x_2 + \dots + x_k = x_0$ |
| topic | Combinatorics 05D10 |
| url | https://arxiv.org/abs/2511.20528 |