Solymosi, József and Swanepoel, Konrad
(2008)
*Elementary incidence theorems for complex numbers and quaternions.*
SIAM Journal on Discrete Mathematics, 22
(3).
pp. 1145-1148.
ISSN 0895-4801

## 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 $\ell$ in the complex affine plane such that $\lvert(A\times B)\cap\ell\rvert=2$. 2. Let $S$ be a finite noncollinear set of points in the complex affine plane. Then there exists a line $\ell$ such that $2\leq \lvert S\cap\ell\rvert \leq 5$. 3. Let $A$ and $B$ be finite sets of at least two quaternions each. Then there exists a line $\ell$ in the quaternionic affine plane such that $2\leq \lvert(A\times B)\cap\ell\rvert \leq 5$. 4. Let $S$ be a finite noncollinear set of points in the quaternionic affine plane. Then there exists a line $\ell$ such that $2\leq \lvert S\cap\ell\rvert \leq 24$.

Item Type: | Article |
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Official URL: | http://www.siam.org/journals/sidma.php |

Additional Information: | © 2009 SIAM |

Library of Congress subject classification: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Rights: | http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx |

Date Deposited: | 09 Oct 2009 09:34 |

URL: | http://eprints.lse.ac.uk/25420/ |

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