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A note on \(n_k\) configurations and theorems in projective space. (English) Zbl 1122.51002
Summary: A method of embedding \(n_k\) 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” \(n_{n-k}\) “theorem” about projective space (over a field or skew-field \(F)\) from any \(n_k\) 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 \(7_3\), Pappus \(9_3\), Desargues \(10_3\) (also in \(3d\)-space), Möbius \(8_4\) (in \(3d\)-space), and the resulting \(7_d\) in \(3d\)-space, \(9_6\) in \(5d\)-space, and \(10_7\) in \(6d\)-space. (The Möbius configuration is self-complementary.) There are some \(n_k\) 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.

MSC:
51A20 Configuration theorems in linear incidence geometry
51A45 Incidence structures embeddable into projective geometries
05B25 Combinatorial aspects of finite geometries
05B35 Combinatorial aspects of matroids and geometric lattices
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