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Small triangle-free configurations of points and lines. (English) Zbl 1109.05029
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 $v\le 18$ 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.
##### MSC:
 05B30 Other designs, configurations 51A20 Configuration theorems (geometry) 51A45 Incidence structures imbeddable into projective geometries
##### Keywords:
geometric realization; Levi graph; collineation