Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]