Elementary incidence theorems for complex numbers and quaternions

Abstract

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.