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New Sufficient Conditions for Linear-Sized Epsilon-Nets and $(p,2)$-Theorems

Chaya Keller, Shakhar Smorodinsky

TL;DR

This work investigates when geometric hypergraphs admit linear-sized piercing sets, focusing on $ε$-nets of size $O(1/ε)$ and linear $(p,2)$-theorems. It establishes sufficient criteria linking Zarankiewicz-type bounds, hereditarily linear Delaunay graphs, and linear/planar supports to these optimal piercing properties, revealing deep connections to graph theory and incidence geometry. Notably, a linear bound in Zarankiewicz-type problems implies a linear $ε$-net theorem for the corresponding neighborhood hypergraphs, and a hereditary linear Delaunay structure for the dual yields an almost linear, then linear, $(p,2)$-theorem for the primal, with concrete improvements for incidence hypergraphs of non-piercing regions. The results unify and extend prior work, provide a coherent framework, and pose multiple open questions for further exploration in Zarankiewicz bounds, shallow cell complexity, and duality of $ε$-net properties.

Abstract

An $ε$-net theorem for a hypergraph upper bounds the minimum size of a vertex set that pierces all $ε$-heavy hyperedges. A $(p,2)$-theorem bounds from above the minimum size of a vertex set that pierces all hyperedges, in terms of the maximum size of a set of pairwise disjoint hyperedges. Numerous works studied $ε$-net theorems and $(p,2)$-theorems that guarantee the existence of small-sized piercing sets. We focus on the question: In which settings the asymptotically smallest possible piercing sets -- i.e., $ε$-nets of size $O(\frac{1}ε)$ and piercing sets of size $O(p)$ in $(p,2)$-theorems, are guaranteed? We obtain several sufficient criteria for the existence of such linear $ε$-net theorems and $(p,2)$-theorems that unveil interesting connections to graph theory and improve and generalize several previous results. Most notably, we exhibit an unexpected relation of $ε$-nets to the classical Zarankiewicz's problem in graph theory. We show that a linear bound in the Zarankiewicz-type problem that asks for the maximum size of a bipartite graph with no copy of $K_{2,t}$, implies a linear $ε$-net theorem for the corresponding neighborhood hypergraph. We also show that hypergraphs with a hereditarily linear-sized Delaunay graph admit an almost linear $(p,2)$-theorem, and deduce that incidence hypergraphs of non-piercing regions in the plane admit a linear $(p,2)$-theorem, significantly improving previous results on such hypergraphs. Our work presents a landscape of sufficient conditions for the existence of linear $ε$-net theorems and $(p,2)$-theorems, with complex interrelations between them. Many of the interrelations are still unknown and call for future research.

New Sufficient Conditions for Linear-Sized Epsilon-Nets and $(p,2)$-Theorems

TL;DR

This work investigates when geometric hypergraphs admit linear-sized piercing sets, focusing on -nets of size and linear -theorems. It establishes sufficient criteria linking Zarankiewicz-type bounds, hereditarily linear Delaunay graphs, and linear/planar supports to these optimal piercing properties, revealing deep connections to graph theory and incidence geometry. Notably, a linear bound in Zarankiewicz-type problems implies a linear -net theorem for the corresponding neighborhood hypergraphs, and a hereditary linear Delaunay structure for the dual yields an almost linear, then linear, -theorem for the primal, with concrete improvements for incidence hypergraphs of non-piercing regions. The results unify and extend prior work, provide a coherent framework, and pose multiple open questions for further exploration in Zarankiewicz bounds, shallow cell complexity, and duality of -net properties.

Abstract

An -net theorem for a hypergraph upper bounds the minimum size of a vertex set that pierces all -heavy hyperedges. A -theorem bounds from above the minimum size of a vertex set that pierces all hyperedges, in terms of the maximum size of a set of pairwise disjoint hyperedges. Numerous works studied -net theorems and -theorems that guarantee the existence of small-sized piercing sets. We focus on the question: In which settings the asymptotically smallest possible piercing sets -- i.e., -nets of size and piercing sets of size in -theorems, are guaranteed? We obtain several sufficient criteria for the existence of such linear -net theorems and -theorems that unveil interesting connections to graph theory and improve and generalize several previous results. Most notably, we exhibit an unexpected relation of -nets to the classical Zarankiewicz's problem in graph theory. We show that a linear bound in the Zarankiewicz-type problem that asks for the maximum size of a bipartite graph with no copy of , implies a linear -net theorem for the corresponding neighborhood hypergraph. We also show that hypergraphs with a hereditarily linear-sized Delaunay graph admit an almost linear -theorem, and deduce that incidence hypergraphs of non-piercing regions in the plane admit a linear -theorem, significantly improving previous results on such hypergraphs. Our work presents a landscape of sufficient conditions for the existence of linear -net theorems and -theorems, with complex interrelations between them. Many of the interrelations are still unknown and call for future research.

Paper Structure

This paper contains 14 sections, 11 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Linear lopsided Zarankiewicz bound implies linear $\epsilon$-net theorem
  5. Hereditarily linear Delaunay graph for the dual hypergraph implies an almost linear $(p,2)$-theorem for the primal
  6. Pseudo-disks, non-piercing regions and linear support
  7. A figure summarizing our results and open problems
  8. From Zarankiewicz's Problem to Linear $\epsilon$-Nets
  9. From a Hereditarily Linear Delaunay Graph to $(p,2)$-Theorems
  10. A linear fractional Helly theorem
  11. An almost linear $(p,2)$-theorem
  12. A linear $(p,2)$-theorem and an application to non-piercing regions
  13. Discussion and Open Problems
  14. Linear support implies bounded VC-dimension

Key Result

Theorem 1.4

Let $c>0$ and $d \in \mathbb{N}$. Let $\mathcal{G}$ be a class of bipartite graphs with VC-dimension $d$ that admits a hereditarily linear lopsided Zarankiewicz theorem, with a constant $c(\mathcal{G})=c$. Then for every $G \in \mathcal{G}$, the hypergraph $H_G$ admits an $\epsilon$-net of size at m

Figures (1)

  • Figure 1: The diagram depicts the relationships between the various problems discussed in this paper. Here, $\mathcal{G}$ is a class of bipartite graphs and $\mathcal{H}=\mathcal{H}_{\mathcal{G}}$ is the class of the corresponding primal neighborhood hypergraphs. Our main results are indicated by bold arrows. The references to the text are as follows: * = Corollary \ref{['cor:3']}, ** = Theorem \ref{['thm:2']}, *** =Theorem \ref{['thm:lopsided->epsnet']}, **** = Proposition \ref{['thm:1']}, ***** = Proposition \ref{['thm:supp->Zaran']}. The normal-line arrows illustrate previously known relations. In particular, 'A' is the main result of PyrgaR08, 'B' is mentioned in Raman2020 and is discussed in Section \ref{['sec:disc']}, 'C' and 'D' are proved in ChanGKS12MustafaDG18 and are discussed in Section \ref{['sec:disc']}, and 'E' follows immediately from the definitions. The dashed arrows denote open problems that are discussed in Section \ref{['sec:disc']}.

Theorems & Definitions (23)

  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Proposition 1.7: A fractional Helly theorem for hypergraphs with a hereditarily linear Delaunay graph
  • Corollary 1.8
  • Proposition 1.9
  • proof
  • proof
  • Lemma 3.1
  • ...and 13 more