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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 ϕ of (v 3 ) into the Euclidean plane, i.e., ϕ is a map such that points are collinear if and only if their images are. The image of ϕ is called a geometric (v 3 ). For triangle-free (v 3 ) with v18 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:
05B30Other designs, configurations
51A20Configuration theorems (geometry)
51A45Incidence structures imbeddable into projective geometries