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Let V be a finite set of points in the plane, not all on one line, and let l be a line that contains at least 2 points of V. We say that l is a k-bisector of V if there are at least k points of V on each one of the two open half-planes bounded by l. For $$k \ge 6$$ k ≥ 6 we construct planar sets of $$2k+4$$ 2 k + 4 points having no k-bisector (this might be best possible). Furthermore,...
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