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Partitioning the \(n\)-space into collinear sets of orthants. (English) Zbl 1062.05110
How many open \(n\)-orthants in \(n\)-dimensional Euclidean space can be intersected by a single straight line? The author proves that the maximum is \(n+1\), and that the maximum is attained for each line in general position (relative to the coordinate hyperplanes). Let \(\kappa(n)\) denote the minimum number of straight lines in general position such that each open \(n\)-orthant is intersected by one of the lines. This number \(\kappa(n)\) is relevant in the context of coverings of hypercubes. It was conjectured that the vertex-set of the \(n\)-dimensional hypercube can be covered by \(\kappa(n)\) \(n\)-stars, where each \(n\)-star consists of a vertex and the \(n\) edges emanating from the vertex. The author disproves the conjecture by showing that \(\kappa(5)=6\), but that six \(5\)-stars are not enough to cover the vertex-set of the \(5\)-dimensional hypercube.
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
52B11 \(n\)-dimensional polytopes
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)