On generalizing the Van der Waerden theorem to some symmetric functions
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914148494868480 |
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| author | Bialostocki, Arie Oles, Vladyslav |
| author_facet | Bialostocki, Arie Oles, Vladyslav |
| contents | Let $n,m$ be positive integers and $c \in \mathbb{Z}_n$, where $\mathbb{Z}_n$ is the ring of integers modulo $n$. We almost complete providing the answer to the following problem, partially solved by N. Alon. Does any infinite sequence over $\mathbb{Z}_n$ contain $m$ same-length consecutive blocks $B_1, \ldots, B_m$ s.t. $\sum B_j + c \prod B_j = 0$ for every $j=1,\ldots,m$ (where $\sum B$ and $\prod B$ denote, respectively, the sum and the product of the elements in block $B$)? In the case of $c=0$, this problem is equivalent to the Van der Waerden theorem. After investigating $B \mapsto \sum B + c\prod B$, we provide other examples of generalizing the Van der Waerden theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07921 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalizing the Van der Waerden theorem to some symmetric functions Bialostocki, Arie Oles, Vladyslav Number Theory Combinatorics Let $n,m$ be positive integers and $c \in \mathbb{Z}_n$, where $\mathbb{Z}_n$ is the ring of integers modulo $n$. We almost complete providing the answer to the following problem, partially solved by N. Alon. Does any infinite sequence over $\mathbb{Z}_n$ contain $m$ same-length consecutive blocks $B_1, \ldots, B_m$ s.t. $\sum B_j + c \prod B_j = 0$ for every $j=1,\ldots,m$ (where $\sum B$ and $\prod B$ denote, respectively, the sum and the product of the elements in block $B$)? In the case of $c=0$, this problem is equivalent to the Van der Waerden theorem. After investigating $B \mapsto \sum B + c\prod B$, we provide other examples of generalizing the Van der Waerden theorem. |
| title | On generalizing the Van der Waerden theorem to some symmetric functions |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2502.07921 |