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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.13726 |
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| _version_ | 1866913274907328512 |
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| author | Montejano, Amanda |
| author_facet | Montejano, Amanda |
| contents | For positive integers $t$ and $n$ let $C_t^n$ be the $n$-cube over $t$ elements, that is, the set of ordered $n$-tuples over the alphabet $\{0,\dots, t-1\}$. We address the question of whether a balanced finite coloring of $C_t^n$ guarantees the presence of a rainbow geometric or combinatorial line. For every even $t\geq 4$ and every $n$, we provide a $\left(\frac{t}{2}\right)^n$--coloring of $C_t^n$ such that all color classes have the same size, and without rainbow combinatorial or geometric lines. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_13726 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rainbow considerations around the Hales-Jewett theorem Montejano, Amanda Combinatorics For positive integers $t$ and $n$ let $C_t^n$ be the $n$-cube over $t$ elements, that is, the set of ordered $n$-tuples over the alphabet $\{0,\dots, t-1\}$. We address the question of whether a balanced finite coloring of $C_t^n$ guarantees the presence of a rainbow geometric or combinatorial line. For every even $t\geq 4$ and every $n$, we provide a $\left(\frac{t}{2}\right)^n$--coloring of $C_t^n$ such that all color classes have the same size, and without rainbow combinatorial or geometric lines. |
| title | Rainbow considerations around the Hales-Jewett theorem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.13726 |