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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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