@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 = "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.",

note = "Japanese Conference on Discrete and Computational Geometry and Graphs : JCDCGG 2015 ; Conference date: 14-09-2015 Through 16-09-2015",

}