Department of Mathematics/IAG, University of Magdeburg, Universitätsplatz 2, D-39106 Magdeburg, Germany, e-mail: bernulf.weissbach@mathematik.uni-magdeburg.de
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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