A generalization of crossing families

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For a set of points in the plane, a crossing family is a set of line segments, each joining two of the points, such that any two line segments cross. We investigate the following generalization of crossing families: a spoke set is a set of lines drawn through a point set such that each unbounded region of the induced line arrangement contains at least one point of the point set. We show that every point set has a spoke set of size √n/8. We also characterize the matchings obtained by selecting exactly one point in each unbounded region and connecting every such point to the point in the antipodal unbounded region.