### Abstract

Considering a variation of the classical Erdos-Szekeres type problems, we count the number of general 4-holes (not necessarily convex, empty 4-gons) in squared Horton sets of size $ntimesn$. Improving on previous upper and lower bounds we show that this number is $n^2log n)$, which constitutes the currently best upper bound on minimizing the number of general $4$-holes for any set of $n$ points in the plane. To obtain the improved bounds, we prove a result of independent interest. We show that $d=1^n d)d^2 = log n)$, where $d)$ is Euler's phi-function, the number of positive integers less than~$d$ which are relatively prime to $d$. This arithmetic function is also called Euler's totient function and plays a role in number theory and cryptography.

Original language | English |
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Title of host publication | Discrete and Computational Geometry and Graphs. JCDCGG 2015. |

Editors | Jin Akiyama, Hiro Ito, Toshinori Sakai, Yushi Uno |

Publisher | Springer, Cham |

Pages | 1-12 |

Number of pages | 12 |

Volume | 9943 |

ISBN (Print) | 978-3-319-48531-7 |

DOIs | |

Publication status | Published - 2016 |

### Publication series

Name | Lecture Notes in Computer Science (LNCS) |
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Publisher | Springer, Cham |

### Fields of Expertise

- Information, Communication & Computing

## Cite this

Aichholzer, O., Hackl, T., Valtr, P., & Vogtenhuber, B. (2016). A Note on the Number of General 4-holes in (Perturbed) Grids. In J. Akiyama, H. Ito, T. Sakai, & Y. Uno (Eds.),

*Discrete and Computational Geometry and Graphs. JCDCGG 2015.*(Vol. 9943, pp. 1-12). (Lecture Notes in Computer Science (LNCS)). Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_1