Elementary incidence theorems for complex numbers and quaternions

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dc.creatorSolymosi, Jozsef
dc.creatorSwanepoel, Konrad J.
dc.date2007-03-13
dc.date.accessioned2026-07-07T12:51:42Z
dc.date.available2026-07-07T12:51:42Z
dc.descriptionWe 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.
dc.description5 pages
dc.identifierhttps://arxiv.org/abs/math/0703372
dc.identifierhttp://arxiv.org/abs/math/0703372
dc.identifierSIAM J. Discrete Math. 22 (2008), 1145--1148.
dc.identifierdoi:10.1137/070685117
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/223076
dc.subjectCombinatorics
dc.subjectMetric Geometry
dc.subject52C10 (Primary) 51A30 (Secondary)
dc.titleElementary incidence theorems for complex numbers and quaternions
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