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

Oswin Aichholzer, Thomas Hackl, Pavel Valtr, Birgit Vogtenhuber

Publikation: Beitrag in Buch/Bericht/KonferenzbandBeitrag in einem Konferenzband

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.
Originalspracheenglisch
TitelDiscrete and Computational Geometry and Graphs. JCDCGG 2015.
Redakteure/-innenJin Akiyama, Hiro Ito, Toshinori Sakai, Yushi Uno
Herausgeber (Verlag)Springer, Cham
Seiten1-12
Seitenumfang12
Band9943
ISBN (Print)978-3-319-48531-7
DOIs
PublikationsstatusVeröffentlicht - 2016
VeranstaltungJapanese Conference on Discrete and Computational Geometry and Graphs: JCDCGG 2015 - Kyoto, Japan
Dauer: 14 Sep 201516 Sep 2015

Publikationsreihe

NameLecture Notes in Computer Science (LNCS)
Herausgeber (Verlag)Springer, Cham

Konferenz

KonferenzJapanese Conference on Discrete and Computational Geometry and Graphs
LandJapan
OrtKyoto
Zeitraum14/09/1516/09/15

Fields of Expertise

  • Information, Communication & Computing

Dieses zitieren