On monochromatic solutions to linear equations over the integers
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914992792535040 |
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| author | Dong, Dingding Mani, Nitya Pham, Huy Tuan Tidor, Jonathan |
| author_facet | Dong, Dingding Mani, Nitya Pham, Huy Tuan Tidor, Jonathan |
| contents | We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of $\{1,\ldots,n\}$. We further study commonness of four-term equations and disprove a conjecture of Costello and Elvin by showing that, unlike over $\mathbb{F}_p$, the four-term equation $x_1 + 2x_2 - x_3 - 2x_4 = 0$ is uncommon over $\{1,\ldots,n\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On monochromatic solutions to linear equations over the integers Dong, Dingding Mani, Nitya Pham, Huy Tuan Tidor, Jonathan Combinatorics 05D40 We study the number of monochromatic solutions to linear equations in a $2$-coloring of $\{1,\ldots,n\}$. We show that any nontrivial linear equation has a constant fraction of solutions that are monochromatic in any $2$-coloring of $\{1,\ldots,n\}$. We further study commonness of four-term equations and disprove a conjecture of Costello and Elvin by showing that, unlike over $\mathbb{F}_p$, the four-term equation $x_1 + 2x_2 - x_3 - 2x_4 = 0$ is uncommon over $\{1,\ldots,n\}$. |
| title | On monochromatic solutions to linear equations over the integers |
| topic | Combinatorics 05D40 |
| url | https://arxiv.org/abs/2410.13758 |