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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.405
URN: urn:nbn:de:0030-drops-50852
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5085/
Green, Ben J.
The Dirac-Motzkin Problem on Ordinary Lines and the Orchard Problem (Invited Talk)
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}
}
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Keywords: |
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combinatorial geometry, incidences |
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Collection: |
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31st International Symposium on Computational Geometry (SoCG 2015) |
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Issue Date: |
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2015 |
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Date of publication: |
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12.06.2015 |
