## Abstract

Given a set system (X, R) such that every pair of sets in R have large symmetric dif-ference, the Shallow Packing Lemma gives an upper bound on |R| as a function of the shallow-cell complexity of R. In this paper, we first present a matching lower bound. Then we give our main theorem, an application of the Shallow Packing Lemma: given a semialgebraic set system (X, R) with shallow-cell complexity ϕ(·, ·) and a parameter ɛ > 0, there exists a collection, called an ɛ-Mnet, consisting of (Formula Presented) subsets of X, each of size (Formula Presented), such that any R ∈ R with | R| ≥ ɛ| X| contains at least one set in this collection. We observe that as an immediate corollary an alternate proof of the optimal ɛ-net bound follows.

Original language | English |
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Pages (from-to) | 756-777 |

Number of pages | 22 |

Journal | Discrete and Computational Geometry |

Volume | 61 |

Issue number | 4 |

DOIs | |

State | Published - 15 Mar 2019 |

Externally published | Yes |

## Keywords

- Epsilon-nets
- Haussler’s Packing Lemma
- Mnets
- Shallow Packing Lemma
- Shallow-cell complexity

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics