A note on multidimensional Ramsey numbers
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910783670059008 |
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| author | Mubayi, Dhruv |
| author_facet | Mubayi, Dhruv |
| contents | Fix integers $d,r\ge 2$ and suppose that the edge set of the $d$-fold Cartesian product of the $N$-clique $K_N^d$ is $r$-colored. We show that there is a copy of $K_n^d$ whose edges in each direction are monochromatic provided $N > 2^{2^{c n^{d-1}}}$, where $c$ depends only on $r$ and $d$. This improves the previous best exponent of $n^d$ proved by Girão, Kronenberg, and Scott while also improving the best known bound due to them for a multidimensional Erd\H os-Szekeres monotone subsequence theorem introduced by Fishburn and Graham. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02389 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on multidimensional Ramsey numbers Mubayi, Dhruv Combinatorics Fix integers $d,r\ge 2$ and suppose that the edge set of the $d$-fold Cartesian product of the $N$-clique $K_N^d$ is $r$-colored. We show that there is a copy of $K_n^d$ whose edges in each direction are monochromatic provided $N > 2^{2^{c n^{d-1}}}$, where $c$ depends only on $r$ and $d$. This improves the previous best exponent of $n^d$ proved by Girão, Kronenberg, and Scott while also improving the best known bound due to them for a multidimensional Erd\H os-Szekeres monotone subsequence theorem introduced by Fishburn and Graham. |
| title | A note on multidimensional Ramsey numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.02389 |