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arXiv · 2509.06403

Maximum in-general-position set in a random subset of $\mathbb{F}^d_q$

Yaobin Chen·Jiaxi Nie·Jing Yu·Wentao Zhang
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Abstract

Let $\alpha(\mathbb{F}_q^{d},p)$ 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 $\alpha(\mathbb{F}_q^{d},p)$ 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.

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Yaobin Chen, Jiaxi Nie, Jing Yu, Wentao Zhang. 2025-09-08. Maximum in-general-position set in a random subset of $\mathbb{F}^d_q$. https://arxiv.org/abs/2509.06403

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