## Abstract

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in R^{d} (vertices with exactly k of the hyperplanes passing below them). This is essentially a dual version of the k-set problem, which, in a primal setting, seeks bounds for the maximum number of k-sets determined by n points in R^{d}, where a k-set is a subset of size k that can be separated from its complement by a hyperplane. The k-set problem is still wide open even in the plane. In three dimensions, the best known upper and lower bounds are, respectively, O(nk^{3}/^{2}) [15] and nk · 2^{Ω(}log k^{)} [19]. In its dual version, the problem can be generalized by replacing hyperplanes by other families of surfaces (or curves in the planes). Reasonably sharp bounds have been obtained for curves in the plane [16, 18], but the known upper bounds are rather weak for more general surfaces, already in three dimensions, except for the case of triangles [1]. The best known general bound, due to Chan [7] is O(n^{2.997}), for families of surfaces that satisfy certain (fairly weak) properties. In this paper we consider the case of pseudoplanes in R^{3} (defined in detail in the introduction), and establish the upper bound O(nk^{5}/^{3}) for the number of k-level vertices in an arrangement of n pseudoplanes. The bound is obtained by establishing suitable (and nontrivial) extensions of dual versions of classical tools that have been used in studying the primal k-set problem, such as the Lovász Lemma and the Crossing Lemma.

Original language | English |
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Title of host publication | 35th International Symposium on Computational Geometry, SoCG 2019 |

Editors | Gill Barequet, Yusu Wang |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771047 |

DOIs | |

State | Published - 1 Jun 2019 |

Event | 35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States Duration: 18 Jun 2019 → 21 Jun 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 129 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 35th International Symposium on Computational Geometry, SoCG 2019 |
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Country/Territory | United States |

City | Portland |

Period | 18/06/19 → 21/06/19 |

### Funding

Funders | Funder number |
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Blavatnik Research Fund in Computer Science | |

German-Israeli Foundation for Scientific Research and Development | G-1367-407.6/2016 |

Israel Science Foundation | 892/13, 260/18 |

Tel Aviv University | |

Israeli Centers for Research Excellence | 4/11 |

## Keywords

- Arrangements
- K-level
- K-sets
- Pseudoplanes
- Three dimensions