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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.13115 |
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| _version_ | 1866911157878521856 |
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| author | Gangopadhyay, Rahul Polyanskii, Alexander Rao, Wei |
| author_facet | Gangopadhyay, Rahul Polyanskii, Alexander Rao, Wei |
| contents | In 2008, Halman showed that for any finite set $P\subset \mathbb R^d$ and any finite family $\mathcal{B}$ of axis-parallel boxes in $\mathbb{R}^d$, if the intersection of $P$ and any subfamily $\mathcal{B}' \subseteq\mathcal{B}$ of size at most $2d$ is non-empty, then the intersection of $P$ and $\mathcal{B}$ is also non-empty. Very recently Edwards and Soberón initiated the study of quantitative colorful version for $2d$ families, $(p,q)$-type variation for $p\geq q\geq d+1$, and other extensions of this Helly-type result by Halman.
In this paper, we study the quantitative colorful Halman problem for $2d-1$ families as well its $(p,q)$-type variation for $p\geq q\geq 2$. Specifically, our main result asserts that for any finite set $P$ and finite families of boxes $\mathcal{B}_1,\dots,\mathcal{B}_{2d-1}$ in $\mathbb R^d$, where $d\geq 2$, if every transversal $\mathcal{B}$ for the families has an intersection $\bigcap \mathcal{B}$ containing at least $n$ points of $P$, then there exist $j\in[2d-1]$ and a subset of $P$ of size at most \[ 2n+\Big\lfloor \frac{n-1}{d \cdot 2^{d-1}} \Big\rfloor, \] such that each box of $\mathcal{B}_j$ contains at least $n$ points of this subset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_13115 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New Helly-type results for discrete boxes: Quantitative colorful and $(p,q)$-variants Gangopadhyay, Rahul Polyanskii, Alexander Rao, Wei Combinatorics In 2008, Halman showed that for any finite set $P\subset \mathbb R^d$ and any finite family $\mathcal{B}$ of axis-parallel boxes in $\mathbb{R}^d$, if the intersection of $P$ and any subfamily $\mathcal{B}' \subseteq\mathcal{B}$ of size at most $2d$ is non-empty, then the intersection of $P$ and $\mathcal{B}$ is also non-empty. Very recently Edwards and Soberón initiated the study of quantitative colorful version for $2d$ families, $(p,q)$-type variation for $p\geq q\geq d+1$, and other extensions of this Helly-type result by Halman. In this paper, we study the quantitative colorful Halman problem for $2d-1$ families as well its $(p,q)$-type variation for $p\geq q\geq 2$. Specifically, our main result asserts that for any finite set $P$ and finite families of boxes $\mathcal{B}_1,\dots,\mathcal{B}_{2d-1}$ in $\mathbb R^d$, where $d\geq 2$, if every transversal $\mathcal{B}$ for the families has an intersection $\bigcap \mathcal{B}$ containing at least $n$ points of $P$, then there exist $j\in[2d-1]$ and a subset of $P$ of size at most \[ 2n+\Big\lfloor \frac{n-1}{d \cdot 2^{d-1}} \Big\rfloor, \] such that each box of $\mathcal{B}_j$ contains at least $n$ points of this subset. |
| title | New Helly-type results for discrete boxes: Quantitative colorful and $(p,q)$-variants |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.13115 |