## Abstract

Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let h
_{5}(n) be the minimum number of 5-holes among all sets of n points in the plane in general position. Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for h
_{5}(n) have been of order Ω(n) and O(n
^{2}), respectively. We show that h
_{5}(n)=Ω(nlog
^{4/5}n), obtaining the first superlinear lower bound on h
_{5}(n). The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line ℓ into two subsets, each of size at least 5 and not in convex position, then ℓ intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted.

Original language | English |
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Article number | 105236 |

Number of pages | 31 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 173 |

DOIs | |

Publication status | Published - Jul 2020 |

Externally published | Yes |

### Bibliographical note

DBLP's bibliographic metadata records provided through http://dblp.org/search/publ/api are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.## Keywords

- Empty k-gon
- Empty pentagon
- Erdös–Szekeres type problem
- Planar point set
- k-Hole