Elementary incidence theorems for complex numbers and quaternions
Loading...
Date
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
Description
We present some elementary ideas to prove the following Sylvester-Gallai type theorems involving incidences between points and lines in the planes over the complex numbers and quaternions.
(1) Let A and B be finite sets of at least two complex numbers each. Then there exists a line l in the complex affine plane such that l intersects AxB in exactly two points.
(2) Let S be a finite noncollinear set of points in the complex affine plane. Then there exists a line l intersecting S in 2, 3, 4 or 5 points.
(3) Let A and B be finite sets of at least two quaternions each. Then there exists a line l in the quaternionic affine plane such that l intersects AxB in 2, 3, 4 or 5 points.
(4) Let S be a finite noncollinear set of points in the quaternionic affine plane. Then there exists a line l intersecting S in at least 2 and at most 24 points.
5 pages
5 pages