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n Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie - 'n Veralgemeende Sylvester-Gallai Stelling : research and review article

Volume 26, Issue 1
  • ISSN : 0254-3486
  • E-ISSN: 2222-4173

Abstract

We give an algorithmic proof for the contrapositive of the following theorem that has recently been proved by the authors:


Let S be a finite set of points in the plane, with each point coloured red, blue or with both colours. Suppose that for any two distinct points and in sharing a colour , there is a third point in which has (inter alia) the colour different from and is collinear with and . Then all the points in S are collinear.
This theorem is a generalization of both the Sylvester-Gallai Theorem and the Motzkin-Rabin Theorem.

Ons gee 'n algoritmiese bewys vir die kontrapositief van die volgende stelling wat onlangs deur die outeurs bewys is:


Laat 'n eindige versameling van punte in die vlak wees, met elke punt rooi, blou of metbeide kleure gekleur. Veronderstel dat daar vir enige twee verskillende punte en in wat 'n kleur k deel, 'n derde punt in is wat (o.a.) die kleur anders as het en wat saamlynig met en is. Dan is al die punte in S saamlynig.
Hierdie stelling is 'n gemeenskaplike veralgemening van die Sylvester-Gallai Stelling en die Motzkin-Rabin Stelling.

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/content/aknat/26/1/EJC20404
2007-03-01
2019-10-23

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