Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 41, No. 2, pp. 417-423 (2000)

Sets with Large Borsuk Number

Bernulf Weissbach

Department of Mathematics/IAG, University of Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany, e-mail:

Abstract: We construct sets in Euclidean spaces of dimension $d={4m-2\choose 2}$, where $m$ is a power of a prime, with the property that they can only be covered with a large number of sets having smaller diameter. Thereby we generalize a result of A.M. Raigorodskii and, in addition, we prove that there exists a counterexample to the so called "Borsuk-conjecture" already in dimension $34\choose 2}-1=560$.

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