### Abstract

Original language | English |
---|---|

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) |
---|---|

Publisher | Springer, Cham |

### Fields of Expertise

- Information, Communication & Computing

### Cite this

*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

**A Note on the Number of General 4-holes in (Perturbed) Grids.** / Aichholzer, Oswin; Hackl, Thomas; Valtr, Pavel; Vogtenhuber, Birgit.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review

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

}

TY - GEN

T1 - A Note on the Number of General 4-holes in (Perturbed) Grids

AU - Aichholzer, Oswin

AU - Hackl, Thomas

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

U2 - https://doi.org/10.1007/978-3-319-48532-4_1

DO - https://doi.org/10.1007/978-3-319-48532-4_1

M3 - Conference contribution

SN - 978-3-319-48531-7

VL - 9943

T3 - Lecture Notes in Computer Science (LNCS)

SP - 1

EP - 12

BT - Discrete and Computational Geometry and Graphs. JCDCGG 2015.

A2 - Akiyama, Jin

A2 - Ito, Hiro

A2 - Sakai, Toshinori

A2 - Uno, Yushi

PB - Springer, Cham

ER -