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DOI: 10.4230/LIPIcs.SOCG.2015.405
URN: urn:nbn:de:0030-drops-50852
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5085/
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Green, Ben J.

The Dirac-Motzkin Problem on Ordinary Lines and the Orchard Problem (Invited Talk)

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Abstract

Suppose you have n points in the plane, not all on a line. A famous theorem of Sylvester-Gallai asserts that there is at least one ordinary line, that is to say a line passing through precisely two of the n points. But how many ordinary lines must there be? It turns out that the answer is at least n/2 (if n is even) and roughly 3n/4 (if n is odd), provided that n is sufficiently large. This resolves a conjecture of Dirac and Motzkin from the 1950s. We will also discuss the classical orchard problem, which asks how to arrange n trees so that there are as many triples of colinear trees as possible, but no four in a line. This is joint work with Terence Tao and reports on the results of [Green and Tao, 2013].

BibTeX - Entry

@InProceedings{green:LIPIcs:2015:5085,
  author =	{Ben J. Green},
  title =	{{The Dirac-Motzkin Problem on Ordinary Lines and the Orchard Problem (Invited Talk)}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{405--405},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5085},
  URN =		{urn:nbn:de:0030-drops-50852},
  doi =		{10.4230/LIPIcs.SOCG.2015.405},
  annote =	{Keywords: combinatorial geometry, incidences}
}

Keywords: combinatorial geometry, incidences
Seminar: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 11.06.2015


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