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Distinguished Math Club Lecture 2004: Directions on Discrete Geometry
Special Events| Speaker: | Prof. Janos Pach, Courant Institute, New York University |
| Location: | 206 Olson Hall |
| Start time: | Fri, Mar 5 2004, 2:10PM |
Description
According to a celebrated theorem of Sylvester and Gallai, any
finite set $S$ of non-collinear points in the plane has two elements
whose connecting line does not pass through any other point in $S$.
Erd\H os noticed that this result immediately implies that any set of
$n$ non-collinear points in the plane determines at least $n$
different connecting lines. Equality is attained if and only if
all but one of the points are on a line. In the same spirit, Scott
posed two similar questions in 1970: (1) Is it true that the
number of different directions assumed by the connecting lines of
$n>3$ non-collinear points in the plane is at least $n-1$? (2) Is
it true that the number of different directions assumed by the
connecting lines of $n>5$ non-coplanar points in $3$-space is at
least $2n-3$?
The first question was answered in the affirmative by Ungar in
1982, using allowable sequences (see {\em Proofs from the Book} by
Aigner and Ziegler). We outline a completely elementary argument
of Pinchasi, Sharir, and the speaker that solves the second
problem of Scott. We also mention several open problems.
There will be a special lunch of the
Undergraduate Math Majors with Prof. Pach.