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Zbl 1109.05029
Boben, Marko; Grünbaum, Branko; Pisanski, Tomaž; Žitnik, Arjana
Small triangle-free configurations of points and lines.
(English)
[J] Discrete Comput. Geom. 35, No. 3, 405-427 (2006). ISSN 0179-5376

A (combinatorial) configuration ($v_{3}$) is a partial linear space, with $v$~points, three points on each line, and three lines passing through each point. For simplicity we identify lines with the set of points incident with them. A ($v_{3}$) is called triangle-free if in any triple of noncollinear points there is a pair of points not joinable by a line. This obviously implies its dual. A geometric realization of a ($v_{3}$) is an injective collineation $\phi$ of ($v_{3}$) into the Euclidean plane, i.e., $\phi$ is a map such that points are collinear if and only if their images are. The image of $\phi$ is called a geometric ($v_{3}$). For triangle-free ($v_{3}$) with $v\le18$ we have up to isomorphism exactly one ($15_{3}$), one ($17_{3}$), and four ($18_{3}$). The authors show that they all posses geometric realizations. Indeed, the maps are given explicitly. Similarly for the unique point transitive ($20_{3}$) and ($21_{3}$). The results are computer-generated.
[Hubert Kiechle (Hamburg)]
MSC 2000:
*05B30 Other designs, configurations
51A20 Configuration theorems (geometry)
51A45 Incidence structures imbeddable into projective geometries

Keywords: geometric realization; Levi graph; collineation

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