No Infinite $(p,q)$-Theorem for Piercing Compact Convex Sets with Lines in $\mathbb{R}^3$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916946551767040 |
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| author | Chakraborty, Sutanoya Ghosh, Arijit |
| author_facet | Chakraborty, Sutanoya Ghosh, Arijit |
| contents | An infinite $(p,q)$-theorem, or an $(\aleph_0,q)$-theorem, involving two families $\mathcal{F}$ and $\mathcal{G}$ of sets, states that if in every infinite subset of $\mathcal{F}$, there are $q$ sets that are intersected by some set in $\mathcal{G}$, then there is a finite set $S_{\mathcal{F}}\subseteq\mathcal{G}$ such that for every $C\in\mathcal{F}$, there is a $B\in S_{\mathcal{F}}$ with $C\cap B\neq\emptyset$. We provide an example demonstrating that there is no $(\aleph_0,q)$-theorem for piercing compact convex sets in $\mathbb{R}^3$ with lines by constructing a family $\mathcal{F}$ of compact convex sets such that it does not have a finite line transversal, but for any $t\in\mathbb{N}$, every infinite subset of $\mathcal{F}$ contains $t$ sets that are pierced by a line. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | No Infinite $(p,q)$-Theorem for Piercing Compact Convex Sets with Lines in $\mathbb{R}^3$ Chakraborty, Sutanoya Ghosh, Arijit Combinatorics Computational Geometry An infinite $(p,q)$-theorem, or an $(\aleph_0,q)$-theorem, involving two families $\mathcal{F}$ and $\mathcal{G}$ of sets, states that if in every infinite subset of $\mathcal{F}$, there are $q$ sets that are intersected by some set in $\mathcal{G}$, then there is a finite set $S_{\mathcal{F}}\subseteq\mathcal{G}$ such that for every $C\in\mathcal{F}$, there is a $B\in S_{\mathcal{F}}$ with $C\cap B\neq\emptyset$. We provide an example demonstrating that there is no $(\aleph_0,q)$-theorem for piercing compact convex sets in $\mathbb{R}^3$ with lines by constructing a family $\mathcal{F}$ of compact convex sets such that it does not have a finite line transversal, but for any $t\in\mathbb{N}$, every infinite subset of $\mathcal{F}$ contains $t$ sets that are pierced by a line. |
| title | No Infinite $(p,q)$-Theorem for Piercing Compact Convex Sets with Lines in $\mathbb{R}^3$ |
| topic | Combinatorics Computational Geometry |
| url | https://arxiv.org/abs/2509.06731 |