Three-color van der Waerden numbers grow super-exponentially
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918535357267968 |
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| author | Fox, Jacob Hunter, Zach |
| author_facet | Fox, Jacob Hunter, Zach |
| contents | For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erdős and Graham on canonical van der Waerden numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02541 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Three-color van der Waerden numbers grow super-exponentially Fox, Jacob Hunter, Zach Combinatorics Number Theory For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erdős and Graham on canonical van der Waerden numbers. |
| title | Three-color van der Waerden numbers grow super-exponentially |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2606.02541 |