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.


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
Full Text: DOI


[1] DOI: 10.1016/S0747-7171(08)80121-6 · Zbl 0766.14043 · doi:10.1016/S0747-7171(08)80121-6
[2] Steinitz, Encyklopädie der Mathematischen Wissenschaften, mit Einschluss ihrer Anwendungen 3 pp 481– (1910) · doi:10.1007/978-3-663-16027-4_7
[3] Möbius, J. f. Math. 3 pp 276– (1828)
[4] Hirschfeld, General Galois geometries (1991) · Zbl 0789.51001
[5] Levi, Finite geometrical systems (1942)
[6] Grünbaum, Bull. Inst. Combin. Appl. 46 pp 15– (2006)
[7] Wallace, Camb. Phil. Proc. XXI pp 348– (1806)
[8] DOI: 10.1016/S0378-3758(99)00124-X · Zbl 0959.51002 · doi:10.1016/S0378-3758(99)00124-X
[9] DOI: 10.1023/A:1005167220050 · Zbl 0937.51001 · doi:10.1023/A:1005167220050
[10] Dembowski, Finite geometries (1968) · doi:10.1007/978-3-642-62012-6
[11] Bokowski, Computational synthetic geometry 1355 (1989) · Zbl 0683.05015 · doi:10.1007/BFb0089253
[12] Baker, Principles of geometry I–IV (1922)
[13] DOI: 10.1016/S0012-365X(96)00327-5 · Zbl 0892.51001 · doi:10.1016/S0012-365X(96)00327-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.