Element-Distinct Solution For Rado's Theorem
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866929608809512960 |
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| author | Yang, Ningyuan Tao, Tianyi |
| author_facet | Yang, Ningyuan Tao, Tianyi |
| contents | In this paper, we present a simplified proof of Rado's Theorem and demonstrate that when an integer matrix $M$ satisfies the column condition and $M\mathbf x=\mathbf 0$ has an element-distinct solution on $\mathbb N$, then under any finite coloring of $\mathbb N$, the equation $M\mathbf x=\mathbf 0$ has a monochromatic element-distinct solution. This gives a positive answer to a problem of Di Nasso in 2016. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17456 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Element-Distinct Solution For Rado's Theorem Yang, Ningyuan Tao, Tianyi Combinatorics In this paper, we present a simplified proof of Rado's Theorem and demonstrate that when an integer matrix $M$ satisfies the column condition and $M\mathbf x=\mathbf 0$ has an element-distinct solution on $\mathbb N$, then under any finite coloring of $\mathbb N$, the equation $M\mathbf x=\mathbf 0$ has a monochromatic element-distinct solution. This gives a positive answer to a problem of Di Nasso in 2016. |
| title | Element-Distinct Solution For Rado's Theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.17456 |