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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.
##### MSC:
 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)