A superlinear improvement on line-free sets in $\mathbb{F}_p^3$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918518240313344 |
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| author | Kovács, Benedek |
| author_facet | Kovács, Benedek |
| contents | Building on an earlier result of the author together with Elsholtz, Führer, Füredi, Pach, Simon and Velich, we present an improved construction for a line-free set in $\mathbb{F}_p^3$, showing that $r_p(\mathbb{F}_p^3)\ge (p-1)^3+\frac18 p^{3/2} - O(p)$ as $p\to \infty$. This results in the first superlinear-term improvement over the standard hypercube construction $\{0,1,\ldots,p-2\}^3$. By taking the complement of our set, we also get a new upper bound of $3p^2-\frac18p^{3/2}+O(p)$ on the smallest size of a $2$-blocking set in the affine geometry $\mathrm{AG}(3,p)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_23437 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A superlinear improvement on line-free sets in $\mathbb{F}_p^3$ Kovács, Benedek Combinatorics 51E21 Building on an earlier result of the author together with Elsholtz, Führer, Füredi, Pach, Simon and Velich, we present an improved construction for a line-free set in $\mathbb{F}_p^3$, showing that $r_p(\mathbb{F}_p^3)\ge (p-1)^3+\frac18 p^{3/2} - O(p)$ as $p\to \infty$. This results in the first superlinear-term improvement over the standard hypercube construction $\{0,1,\ldots,p-2\}^3$. By taking the complement of our set, we also get a new upper bound of $3p^2-\frac18p^{3/2}+O(p)$ on the smallest size of a $2$-blocking set in the affine geometry $\mathrm{AG}(3,p)$. |
| title | A superlinear improvement on line-free sets in $\mathbb{F}_p^3$ |
| topic | Combinatorics 51E21 |
| url | https://arxiv.org/abs/2605.23437 |