# Ordinary lines in space

## Frank de Zeeuw, Baruch College/CUNY

## Date and time: 2pm, Friday, October 5, 2018

## Place: CUNY Graduate Center, Rm 4419

The Sylvester-Gallai theorem states that if a finite set of points in
the real plane is not contained in a line, then it spans at least one
ordinary line, i.e., a line containing exactly two of the points. In
fact, by a result of Green and Tao, any finite point set in the real
plane spans a linear number of ordinary lines, and that is best
possible because there are point sets on cubic curves that determine
only a linear number of ordinary lines.
One can consider the same question in three-dimensional space. By
projection, the results in the plane hold word-for-word in space, but
the known constructions with a linear number of ordinary lines are
contained in a plane. I will show that if one assumes that the point
set does not have too many points on a plane, then it spans a
quadratic number of ordinary lines. More precisely, for any $a\lt1$
there is a $c\gt0$ such that if we have $n$ points in real space with at
most $an$ points on a plane, then there are at least $cn^2$ ordinary
lines. The proof uses projection and Beck’s theorem.