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The Dirac-Motzkin Problem on Ordinary Lines and the Orchard Problem (Invited Talk) Green, Ben J. combinatorial geometry incidences 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]. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Ben J. Green 2015 Is Part Of LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SOCG.2015.405 urn:nbn:de:0030-drops-50852 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.405 eng https://creativecommons.org/licenses/by/3.0/legalcode