On the number of ordinary circles

Hossein Nassajian Mojarrad; Frank de Zeeuw

On the number of ordinary circles

Combinatorics 2016-05-05 v2 Metric Geometry

Authors: Hossein Nassajian Mojarrad , Frank de Zeeuw

Abstract

We prove that any $n$ points in $\mathbb{R}^2$ , not all on a line or circle, determine at least $\frac{1}{4}n^2-O(n)$ ordinary circles (circles containing exactly three of the $n$ points). The main term of this bound is best possible for even $n$ . Our proof relies on a recent result of Green and Tao on ordinary lines.

Keywords

ordinary circles combinatorial geometry point configurations Green-Tao theorem ordinary lines extremal combinatorics

Cite

@article{arxiv.1412.8314,
  title  = {On the number of ordinary circles},
  author = {Hossein Nassajian Mojarrad and Frank de Zeeuw},
  journal= {arXiv preprint arXiv:1412.8314},
  year   = {2016}
}

Comments

v2: The previous version had a mistake that turned out to be fatal to the proof. The new version has a different proof of the same result, based on the same approach, but requiring a much more detailed analysis

On the number of ordinary circles

Abstract

Keywords

Cite

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