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A note on piercing discrete rectangles

Wei Rao

Abstract

In 2008, Halman proved a discrete Helly-type theorem for axis-parallel boxes in $\mathbb R^d$. Very recently, this result was extended to the $(p,q)$ setting with $p \geq q \geq d+1$ by Edwards and Soberón, and subsequently to the case $p \geq q \geq 2$ by Gangopadhyay, Polyanskii, and the author of this paper. In this paper, we obtain improved bounds for the $(p,q)$ problem in the case $q=2$ and $d=2$. More precisely, our main result asserts that for any integer $p \geq 2$, any set $P \subseteq \mathbb R^2$, and any finite family $\mathcal B$ of axis-parallel rectangles in $\mathbb R^2$ such that every rectangle contains a point of $P$, if among every $p$ rectangles there exist two whose intersection contains a point of $P$, then there exists a subset $S \subseteq P$ of size at most $O\!\bigl( (p \log \log p)^2 \bigr)$ such that every rectangle contains a point of $S$. Moreover, when $p=2$, the size of $S$ can be bounded by $8$.

A note on piercing discrete rectangles

Abstract

In 2008, Halman proved a discrete Helly-type theorem for axis-parallel boxes in . Very recently, this result was extended to the setting with by Edwards and Soberón, and subsequently to the case by Gangopadhyay, Polyanskii, and the author of this paper. In this paper, we obtain improved bounds for the problem in the case and . More precisely, our main result asserts that for any integer , any set , and any finite family of axis-parallel rectangles in such that every rectangle contains a point of , if among every rectangles there exist two whose intersection contains a point of , then there exists a subset of size at most such that every rectangle contains a point of . Moreover, when , the size of can be bounded by .

Paper Structure

This paper contains 4 sections, 11 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of Theorem \ref{['theorem: N(2,2,2)']}
  4. Proof of Theorem \ref{['theorem: N(p,2,2)']}

Key Result

Theorem 1.1

Let $d$ be a positive integer. Let $P$ be a finite set in $\mathbb{R}^d$, and let $\mathcal{B}$ be a finite family of boxes in $\mathbb{R}^d$. If for every subfamily $\mathcal{B}' \subseteq \mathcal{B}$ of size at most $2d$ the trace $\mathcal{B}'|_P$ is intersecting, then $\mathcal{B}|_P$ is also i

Figures (3)

  • Figure 1: Illustration of set $S_1$ and proof of Claim \ref{['claim: nonempty components']}
  • Figure 2: Illustration of reduction
  • Figure 3: Illustration of the topological argument. In the figure, we consider the point $q=\bigl((\tfrac{1}{2},\tfrac{1}{2}),(\tfrac{1}{2},\tfrac{1}{2})\bigr)$. Moreover, we have $q \in B_{v_1}, B_{v_2}, B_{v_3}$, where $v_1 = ((1,0),(1,0)), v_2 = ((0,1),(1,0)), v_3 = ((0,1),(0,1))$, due to the existence of corresponding sets $I_{B_1}, I_{B_2}, I_{B_3} \in \mathcal{I}$, respectively.

Theorems & Definitions (22)

  • Theorem 1.1: Halman’s theorem; Theorem 2.10 in Hal
  • Theorem 1.2: Theorem 5 in gangopadhyay2025new
  • Theorem 1.3: Theorem 4.1 in tomon2023lower
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1: Theorem 1.4 in kaiser1997transversals
  • Theorem 2.2: Theorem 2 in komiya1994simple
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['theorem: N(2,2,2) in a specific form']}
  • Claim 3.2
  • ...and 12 more