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

Oswin Aichholzer, Thomas Hackl, Pavel Valtr, Birgit Vogtenhuber

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

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 languageEnglish
Title of host publicationDiscrete and Computational Geometry and Graphs. JCDCGG 2015.
EditorsJin Akiyama, Hiro Ito, Toshinori Sakai, Yushi Uno
PublisherSpringer, Cham
Pages1-12
Number of pages12
Volume9943
ISBN (Print)978-3-319-48531-7
DOIs
Publication statusPublished - 2016

Publication series

NameLecture Notes in Computer Science (LNCS)
PublisherSpringer, 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

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

Discrete and Computational Geometry and Graphs. JCDCGG 2015.. ed. / Jin Akiyama; Hiro Ito; Toshinori Sakai; Yushi Uno. Vol. 9943 Springer, Cham, 2016. p. 1-12 (Lecture Notes in Computer Science (LNCS)).

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

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, Lecture Notes in Computer Science (LNCS), Springer, Cham, pp. 1-12. https://doi.org/10.1007/978-3-319-48532-4_1
Aichholzer O, Hackl T, Valtr P, Vogtenhuber B. A Note on the Number of General 4-holes in (Perturbed) Grids. In Akiyama J, Ito H, Sakai T, Uno Y, editors, Discrete and Computational Geometry and Graphs. JCDCGG 2015.. Vol. 9943. Springer, Cham. 2016. p. 1-12. (Lecture Notes in Computer Science (LNCS)). https://doi.org/10.1007/978-3-319-48532-4_1
Aichholzer, Oswin ; Hackl, Thomas ; Valtr, Pavel ; Vogtenhuber, Birgit. / A Note on the Number of General 4-holes in (Perturbed) Grids. Discrete and Computational Geometry and Graphs. JCDCGG 2015.. editor / Jin Akiyama ; Hiro Ito ; Toshinori Sakai ; Yushi Uno. Vol. 9943 Springer, Cham, 2016. pp. 1-12 (Lecture Notes in Computer Science (LNCS)).
@inproceedings{9cb8e981cfa3448cad0d1a356fb3963e,
title = "A Note on the Number of General 4-holes in (Perturbed) Grids",
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.",
author = "Oswin Aichholzer and Thomas Hackl and Pavel Valtr and Birgit Vogtenhuber",
year = "2016",
doi = "https://doi.org/10.1007/978-3-319-48532-4_1",
language = "English",
isbn = "978-3-319-48531-7",
volume = "9943",
series = "Lecture Notes in Computer Science (LNCS)",
publisher = "Springer, Cham",
pages = "1--12",
editor = "Jin Akiyama and Hiro Ito and Toshinori Sakai and Yushi Uno",
booktitle = "Discrete and Computational Geometry and Graphs. JCDCGG 2015.",

}

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 -