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Stabbing boxes with finitely many axis-parallel lines and flats

Sutanoya Chakraborty, Arijit Ghosh, Soumi Nandi

TL;DR

This work establishes an infinite-variant Hadwiger–Debrunner result for axis-parallel boxes: if an infinite family of boxes in $\mathbb{R}^d$ satisfies the $(\aleph_{0},2)$-property for axis-parallel $k$-flats with $0\le k<d$, then a finite-size axis-parallel $k$-transversal exists. The authors prove the main theorem via induction on the ambient dimension, first solving the hyperplane case ($k=d-1$) and then lifting to general $k$ through a dimension-reduction argument, employing strategic constructions of box sequences that resist simultaneous piercing by a single axis-parallel $k$-flat. They explore colorful extensions, showing that a strictly colorful generalization is impossible in general, but obtain a meaningful colorful theorem: if a sequence of box families satisfies the colorful $(\aleph_{0},2)$-property for axis-parallel $k$-flats, then some subfamily admits a finite-size transversal, with a stronger multi-family version also available. These results advance transversal theory for axis-parallel geometric objects and connect to broader Hadwiger–Debrunner-type questions in discrete and computational geometry.

Abstract

In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in $\mathbb{R}^{d}$ to be pierceable by finitely many axis-parallel $k$-flats, where $0 \leq k < d$. We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner $(p, q)$-problem.

Stabbing boxes with finitely many axis-parallel lines and flats

TL;DR

This work establishes an infinite-variant Hadwiger–Debrunner result for axis-parallel boxes: if an infinite family of boxes in satisfies the -property for axis-parallel -flats with , then a finite-size axis-parallel -transversal exists. The authors prove the main theorem via induction on the ambient dimension, first solving the hyperplane case () and then lifting to general through a dimension-reduction argument, employing strategic constructions of box sequences that resist simultaneous piercing by a single axis-parallel -flat. They explore colorful extensions, showing that a strictly colorful generalization is impossible in general, but obtain a meaningful colorful theorem: if a sequence of box families satisfies the colorful -property for axis-parallel -flats, then some subfamily admits a finite-size transversal, with a stronger multi-family version also available. These results advance transversal theory for axis-parallel geometric objects and connect to broader Hadwiger–Debrunner-type questions in discrete and computational geometry.

Abstract

In this short note, we provide the necessary and sufficient condition for an infinite collection of axis-parallel boxes in to be pierceable by finitely many axis-parallel -flats, where . We also consider colorful generalizations of the above result and establish their feasibility. The problem considered in this paper is an infinite variant of the Hadwiger-Debrunner -problem.
Paper Structure (8 sections, 21 theorems, 31 equations, 2 figures)

This paper contains 8 sections, 21 theorems, 31 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our results:
  3. Notations:
  4. Proof of Theorem \ref{['thm-main-theorem']}
  5. Case-I:
  6. Case-II:
  7. Feasibilty of different colorful extensions
  8. Arguments from Chakraborty et al. chakraborty2024heterochromatic

Key Result

Theorem 1.1

Suppose $\mathcal{F}_1,\dots,\mathcal{F}_{d+1}$ are non-empty finite families of convex sets in $\mathbb{R}^d$ such that for all $\left( C_{1}, \dots, C_{d+1}\right) \in \mathcal{F}_{1} \times \dots \times \mathcal{F}_{d+1}$ we have $\bigcap_{i \in [d+1]} C_{i} \neq \emptyset$. Then there exists an

Figures (2)

  • Figure 1: $\{B_{1,1}, B_{2,3},B_{4,4}, B_{5,3},\dots\}$ is a colorful sequence with respect to $\{\mathcal{F}_n\}_{n\in\mathbb{N}}$, where $\mathcal{F}_n=\{B_{n,m}\;|\;m\in\mathbb{N}\}$
  • Figure 2: $\{B_{1,1}, B_{2,3}, B_{3,1}, B_{4,4}, B_{5,3}, B_{6,3}, \dots\}$ is a strongly colorful sequence with respect to $\{\mathcal{F}_n\}_{n\in\mathbb{N}}$, where $\mathcal{F}_n=\{B_{n,m}\;|\;m\in\mathbb{N}\}$

Theorems & Definitions (27)

  • Theorem 1.1: Bárány Barany82: Colorful Helly Theorem
  • Theorem 1.2: Hadwiger and Debrunner HadwigerD57
  • Theorem 1.3: Alon and Kleitman AlonK92: $(p,q)$-Theorem
  • Theorem 1.4: Alon and Kalai AlonK95: $(p,q)$-Theorem for hyperplane transversals
  • Theorem 1.5: Keller and Perles KellerP22: $(\aleph_{0},k+2)$-Theorem for $k$-transversals
  • Theorem 1.6: Keller and Perles KellerP22: $(\aleph_{0},2)$-Theorem for point transversals
  • Theorem 1.7: Keller and Perles KellerP23: $(\aleph_{0},k+2)$-Theorem for near balls
  • Corollary 1.8: Keller and Perles KellerP23: $(\aleph_{0},k+2)$-Theorem for closed balls
  • Theorem 1.9: $(\aleph_{0},2)$-Theorem for boxes and axis-parallel $k$-flats
  • Theorem 1.10: Jung and Pálvölgyi JungP2023: Convex sets and hyperplane transversals
  • ...and 17 more