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.
|Name||Lecture Notes in Computer Science (LNCS)|
|Herausgeber (Verlag)||Springer, Cham|
|Konferenz||Japanese Conference on Discrete and Computational Geometry and Graphs|
|Zeitraum||14/09/15 → 16/09/15|
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