Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, June 21, 2016, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Patrick Schnider

A Generalization of Crossing Families

Given a complete geometric graph, a set of pairwise crossing edges is called a crossing family. A result from 1990 states that every complete geometric graph on n vertices contains a crossing family of size \sqrt{n/12}. In some applications where this result is used, it is however not necessary that the edges actually cross, but only that their endpoints lie "niecly" in some sense. This can be made more precise by the definition of spoke sets: A spoke set is a set of lines through a point set such that in each unbounded region of the induced line arrangement there is at least one point of the point set. It is not to hard to see that this is a generalization of crossing families, in the sense that each crossing family of size k induces a spoke set of size k. We show that every point set contains a spoke set of size \sqrt{n/8}.

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