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].