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A note on Nk configurations and theorems in projective space

Published online by Cambridge University Press:  17 April 2009

David G. Glynn
David G. Glynn
Affiliation:
School of Mathematical Sciences, University of Adelaide, Adelaide SA 5005, Australia e-mail: david.glynn@adelaide.edu.au, dglynn@mac.com
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A method of embedding nk configurations into projective space of k–1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a “complementary” “theorem” about projective space (over a field or skew-field F) from any nk theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues 103 (also in 3d-space), Möbius 84 (in 3d-space), and the resulting 74 in 3d-space, 96 in 5d-space, and 107 in 6d-space. (The Möbius configuration is self-complementary.) There are some nk configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 1 , August 2007 , pp. 15 - 31
Copyright
Copyright © Australian Mathematical Society 2007

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