Elementary Incidence Theorems for Complex Numbers and Quaternions
نویسندگان
چکیده
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 |(A×B) ∩ l| = 2. (2) Let S be a finite noncollinear set of points in the complex affine plane. Then there exists a line l such that 2 ≤ |S ∩ l| ≤ 5. (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 2 ≤ |(A×B) ∩ l| ≤ 5. (4) Let S be a finite noncollinear set of points in the quaternionic affine plane. Then there exists a line l such that 2 ≤ |S ∩ l| ≤ 24.
منابع مشابه
Sylvester-Gallai Theorems for Complex Numbers and Quaternions
A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai Theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a d...
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عنوان ژورنال:
- SIAM J. Discrete Math.
دوره 22 شماره
صفحات -
تاریخ انتشار 2008