Largest dyadic dual VC-dimension of non-piercing families
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909650168840192 |
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| author | Huang, Xinqi Qi, Yuzhen Rong, Mingyuan Xu, Zixiang |
| author_facet | Huang, Xinqi Qi, Yuzhen Rong, Mingyuan Xu, Zixiang |
| contents | The dyadic dual VC-dimension of a set system \( \mathcal{F} \) is the largest integer \( \ell \) such that there exist \( \ell \) sets \( F_1, F_{2}, \dots, F_\ell \in \mathcal{F} \), where every pair \( \{i, j\} \in \binom{[\ell]}{2} \) is witnessed by an element \( a_{i,j} \in F_i \cap F_j \) that does not belong to any other set \( F_k \) with \( k \in [\ell] \setminus \{i, j\} \). In this paper, we determine the largest dyadic dual VC-dimension of a non-piercing family is exactly $4$, providing a rare example where the maximum of this parameter can be determined for a natural family arising from geometry. As an application, we give a short and direct proof that the transversal number \( τ(\mathcal{F}) \) of any non-piercing family is at most \(Cν(\mathcal{F})^9 \), where \( ν(\mathcal{F}) \) is the matching number and $C$ is a constant. This improves a recent result of Pálvölgyi and Zólomy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_13606 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Largest dyadic dual VC-dimension of non-piercing families Huang, Xinqi Qi, Yuzhen Rong, Mingyuan Xu, Zixiang Combinatorics Computational Geometry 52A35, 52C45 The dyadic dual VC-dimension of a set system \( \mathcal{F} \) is the largest integer \( \ell \) such that there exist \( \ell \) sets \( F_1, F_{2}, \dots, F_\ell \in \mathcal{F} \), where every pair \( \{i, j\} \in \binom{[\ell]}{2} \) is witnessed by an element \( a_{i,j} \in F_i \cap F_j \) that does not belong to any other set \( F_k \) with \( k \in [\ell] \setminus \{i, j\} \). In this paper, we determine the largest dyadic dual VC-dimension of a non-piercing family is exactly $4$, providing a rare example where the maximum of this parameter can be determined for a natural family arising from geometry. As an application, we give a short and direct proof that the transversal number \( τ(\mathcal{F}) \) of any non-piercing family is at most \(Cν(\mathcal{F})^9 \), where \( ν(\mathcal{F}) \) is the matching number and $C$ is a constant. This improves a recent result of Pálvölgyi and Zólomy. |
| title | Largest dyadic dual VC-dimension of non-piercing families |
| topic | Combinatorics Computational Geometry 52A35, 52C45 |
| url | https://arxiv.org/abs/2506.13606 |