## DIMACS TR: 2002-19

## Sylvester-Gallai theorem and metric betweenness

### Authors: Vasek Chvatal

**
ABSTRACT
**

Sylvester conjectured in 1893 and Gallai proved some forty
years later that every finite set S of points in the plane includes
two points such that the line passing through them includes either no
other point of S or all other points of S. There are several ways of
extending the notion of lines from Euclidean spaces to arbitrary
metric spaces. We present one of them and conjecture that, with lines
in metric spaces defined in this way, the Sylvester-Gallai theorem
generalizes as follows: in every finite metric space, there is a line
consisting of either two points or all the points of the space. Then
we present slight evidence in support of this rash conjecture and
finally we discuss the underlying ternary relation of metric
betweenness.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/2002/2002-19.ps.gz

DIMACS Home Page