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arXiv · 1409.0081

On $k$-Gons and $k$-Holes in Point Sets

Oswin Aichholzer·Ruy Fabila-Monroy·Hernán González-Aguilar·Thomas Hackl·Marco A. Heredia·Clemens Huemer·Jorge Urrutia·Pavel Valtr·Birgit Vogtenhuber
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Abstract

We consider a variation of the classical Erdős-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position.

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Oswin Aichholzer, Ruy Fabila-Monroy, Hernán González-Aguilar, Thomas Hackl, Marco A. Heredia, Clemens Huemer, Jorge Urrutia, Pavel Valtr, Birgit Vogtenhuber. 2014-08-30. On $k$-Gons and $k$-Holes in Point Sets. https://arxiv.org/abs/1409.0081

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