# 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.