Extraction Theorems With Small Extraction Numbers

TL;DR

Addresses bounding extraction numbers for geometric covering problems by linking extraction to proper coloring of the geometric hypergraph $H(\mathcal{O})$. The main method shows that if $H(\mathcal{O})$ admits a polynomial-time proper $\kappa$-coloring, then an extraction theorem with extraction number $\kappa$ holds. The paper derives tight bounds for four restricted classes: intervals (extraction number $\alpha=2$), axis-parallel segments ($\alpha=4$), axis-parallel rays (types 2,3,4 yielding extraction numbers $2,3,4$), and octants in 3D ($\alpha=4$), with polynomial-time constructions. It also discusses the general applicability of the framework and outlines future directions for other object families such as unit disks in $\mathbb{R}^3$ and rectangles in $\mathbb{R}^2$.

Abstract

In this work, we develop Extraction Theorems for classes of geometric objects with small extraction numbers. These classes include intervals, axis-parallel segments, axis-parallel rays, and octants. We investigate these classes of objects and prove small bounds on the extraction numbers. The tightness of these bounds is demonstrated by examples with matching lower bounds.

Figures (3)

• Figure 1: Configuration reproduced from coaduroindependence demonstrating tightness of extraction number
• Figure 2: Figure showing a lower bound instance for type 3 rays.
• Figure 3: Configuration of projection of octants demonstrating extraction number $\geq 4$

Theorems & Definitions (22)

• Theorem 1: Extraction Theorem for Disks
• Proposition 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 2
• proof
• Theorem 3