On the maximum size of ultrametric orthogonal sets over discrete valued fields
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929471102124032 |
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| author | Aranov, Noy Soffer Behajaina, Angelot |
| author_facet | Aranov, Noy Soffer Behajaina, Angelot |
| contents | Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $\mathbb{R}^n$, there is a well-studied notion of "ultrametric orthogonality" in $\mathcal{K}^n$. In this paper, motivated by a question of Erd{ő}s in the real case, given integers $k \geq \ell \geq 2$, we investigate the maximum size of a subset $S \subseteq \mathcal{K}^n \setminus\{{\bf 0}\}$ satisfying the following property: for any $E \subseteq S$ of size $k$, there exists $F \subseteq E$ of size $\ell$ such that any two distinct vectors in $F$ are orthogonal. Other variants of this property are also studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12019 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the maximum size of ultrametric orthogonal sets over discrete valued fields Aranov, Noy Soffer Behajaina, Angelot Number Theory Commutative Algebra Combinatorics 05D99, 11T30 Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $\mathbb{R}^n$, there is a well-studied notion of "ultrametric orthogonality" in $\mathcal{K}^n$. In this paper, motivated by a question of Erd{ő}s in the real case, given integers $k \geq \ell \geq 2$, we investigate the maximum size of a subset $S \subseteq \mathcal{K}^n \setminus\{{\bf 0}\}$ satisfying the following property: for any $E \subseteq S$ of size $k$, there exists $F \subseteq E$ of size $\ell$ such that any two distinct vectors in $F$ are orthogonal. Other variants of this property are also studied. |
| title | On the maximum size of ultrametric orthogonal sets over discrete valued fields |
| topic | Number Theory Commutative Algebra Combinatorics 05D99, 11T30 |
| url | https://arxiv.org/abs/2408.12019 |