The Sylvester-Gallai theorem, colourings and algebra |
| |
Authors: | Lou M Pretorius |
| |
Institution: | a Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa b Department of Mathematical Sciences, University of South Africa, PO Box 392, UNISA 0003, South Africa |
| |
Abstract: | Our point of departure is the following simple common generalisation of the Sylvester-Gallai theorem and the Motzkin-Rabin theorem: LetSbe a finite set of points in the plane, with each point coloured red or blue or with both colours. Suppose that for any two distinct pointsA,B∈Ssharing a colour there is a third pointC∈S, of the other colour, collinear withAandB. Then all the points inSare collinear. We define a chromatic geometry to be a simple matroid for which each point is coloured red or blue or with both colours, such that for any two distinct points A,B∈S sharing a colour there is a third point C∈S, of the other colour, collinear with A and B. This is a common generalisation of proper finite linear spaces and properly two-coloured finite linear spaces, with many known properties of both generalising as well. One such property is Kelly’s complex Sylvester-Gallai theorem. We also consider embeddings of chromatic geometries in Desarguesian projective spaces. We prove a lower bound of 51 for the number of points in a three-dimensional chromatic geometry in projective space over the quaternions. Finally, we suggest an elementary approach to the corollary of an inequality of Hirzebruch used by Kelly in his proof of the complex Sylvester-Gallai theorem. |
| |
Keywords: | Sylvester-Gallai theorem Motzkin-Rabin theorem Two-colouring of a finite linear space Proper finite linear space Combinatorial geometry Finite geometry |
本文献已被 ScienceDirect 等数据库收录! |
|