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 contribution

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