Variations on a theme of Sylvester

Noam Elkies
Harvard University

J.J. Sylvester asked in 1893 whether for every finite set S of points in the real plane, not all on a line, there must be a line that meets S in exactly 2 points. Various proofs are known, the first given by Gallai in 1933 (whence the result is known as the Sylvester-Gallai theorem).
But while the original question is now settled, it suggests natural variations in several directions, most of which remain open.
We describe a few of these directions here: replacing real by complex or quaternion geometry (which requires raising the dimension as well); asking how many lines can meet a set S of given size in
3 points (or 4, or 5, or ...); and generalizing from a single set S to several in the same real plane. We hope that the recent methods in incidence geometry that are the main topic of this workshop will also be relevant to some of these Sylvester-like problems.

Back to Workshop IV: Finding Algebraic Structures in Extremal Combinatorial Configurations