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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.06403 |
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| _version_ | 1866912576187662336 |
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| author | Chen, Yaobin Nie, Jiaxi Yu, Jing Zhang, Wentao |
| author_facet | Chen, Yaobin Nie, Jiaxi Yu, Jing Zhang, Wentao |
| contents | Let $α(\mathbb{F}_q^{d},p)$ be the maximum possible size of a point set in general position in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $α(\mathbb{F}_q^{d},p)$ up to a polylogarithmic factor by proving the balanced supersaturation conjecture of Balogh and Luo. Our result also resolves a conjecture implicitly posed by the first author, Liu, the second author and Zeng. In the course of our proof, we establish a lemma that demonstrates a ``structure vs. randomness'' phenomenon for point sets in finite-field linear spaces, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximum in-general-position set in a random subset of $\mathbb{F}^d_q$ Chen, Yaobin Nie, Jiaxi Yu, Jing Zhang, Wentao Combinatorics Let $α(\mathbb{F}_q^{d},p)$ be the maximum possible size of a point set in general position in a $p$-random subset of $\mathbb{F}_q^d$. We determine the order of magnitude of $α(\mathbb{F}_q^{d},p)$ up to a polylogarithmic factor by proving the balanced supersaturation conjecture of Balogh and Luo. Our result also resolves a conjecture implicitly posed by the first author, Liu, the second author and Zeng. In the course of our proof, we establish a lemma that demonstrates a ``structure vs. randomness'' phenomenon for point sets in finite-field linear spaces, which may be of independent interest. |
| title | Maximum in-general-position set in a random subset of $\mathbb{F}^d_q$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.06403 |