Large grid subsets without many cospherical points
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908416527564800 |
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| author | Dong, Zichao Xu, Zijian |
| author_facet | Dong, Zichao Xu, Zijian |
| contents | Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the previously best known lower bound of $n/4$ toward the Erdős--Purdy problem due to Thiele in 1995. For $d \ge 3$, this improves the recent $Ω\bigl( n^{\frac{3}{d+1}-o(1)} \bigr)$ bound due to Suk and White, confirming their conjectured $Ω\bigl( n^{\frac{d}{d+1}} \bigr)$ bound in a strong sense, and asymptotically resolves the generalized Erdős--Purdy problem posed by Brass, Moser, and Pach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_18113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large grid subsets without many cospherical points Dong, Zichao Xu, Zijian Combinatorics Algebraic Geometry 52C35, 52C10, 05D40 Motivated by intuitions from projective algebraic geometry, we provide a novel construction of subsets of the $d$-dimensional grid $[n]^d$ of size $n - o(n)$ with no $d + 2$ points on a sphere or a hyperplane. For $d = 2$, this improves the previously best known lower bound of $n/4$ toward the Erdős--Purdy problem due to Thiele in 1995. For $d \ge 3$, this improves the recent $Ω\bigl( n^{\frac{3}{d+1}-o(1)} \bigr)$ bound due to Suk and White, confirming their conjectured $Ω\bigl( n^{\frac{d}{d+1}} \bigr)$ bound in a strong sense, and asymptotically resolves the generalized Erdős--Purdy problem posed by Brass, Moser, and Pach. |
| title | Large grid subsets without many cospherical points |
| topic | Combinatorics Algebraic Geometry 52C35, 52C10, 05D40 |
| url | https://arxiv.org/abs/2506.18113 |