An Exposition of the Green-Tao Paper “On Sets Defining Few Ordinary Lines”

Jon Lenchner, IBM T.J. Watson Research Center

October 6, 2015

Given a finite collection of points in the plane, a line through two of the points that does not pass through any of the other points in the collection is called “ordinary.” Sylvester conjectured and Gallai first proved that any collection of N not-all-collinear points must determine at least one ordinary line. A famous conjecture attributed to Dirac and Motzkin that stood open for over 60 years, asserted that for sufficiently large N, any collection of N points in the plane, not all lying on a line, must give rise to at least N/2 distinct ordinary lines. Two years ago, Ben Green and Terence Tao proved this result using methods of algebraic geometry. Their result also gives an asymptotic solution to the so-called orchard-planting problem. In this talk I will give an outline of the Green-Tao proof, describe one or two of he most important arguments, and also provide an overview of the necessary background in algebraic geometry needed to understand the proof.