On Polynomial Extensions of van der Waerden's Theorem and its Applications
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909711610150912 |
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| author | Goswami, Sayan |
| author_facet | Goswami, Sayan |
| contents | In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an $IP_r^\star$-strengthened version of the polynomial van der Waerden theorem, where the recurrence set is guaranteed to be large in a precise combinatorial sense. As applications, we obtain new monochromatic polynomial configurations in both additive and multiplicative settings, including refined results over sum subsystems of IP-sets. Additionally, we prove exponential monochromatic patterns are abundant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10550 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Polynomial Extensions of van der Waerden's Theorem and its Applications Goswami, Sayan Combinatorics 05D10, 05C55, 22A15, 54D35 In this article, we investigate polynomial generalizations of the van der Waerden theorem with a focus on largeness properties of recurrence patterns. We prove an $IP_r^\star$-strengthened version of the polynomial van der Waerden theorem, where the recurrence set is guaranteed to be large in a precise combinatorial sense. As applications, we obtain new monochromatic polynomial configurations in both additive and multiplicative settings, including refined results over sum subsystems of IP-sets. Additionally, we prove exponential monochromatic patterns are abundant. |
| title | On Polynomial Extensions of van der Waerden's Theorem and its Applications |
| topic | Combinatorics 05D10, 05C55, 22A15, 54D35 |
| url | https://arxiv.org/abs/2401.10550 |