Rectilinear crossing number of the complete graph

Determine, for each positive integer n, the minimum number of pairs of crossing straight-line segments in a drawing of the complete graph K_n formed by joining every pair of points in an n-point set in the plane.

Background

The paper introduces order types through orientation predicates on triples of points and notes that many geometric quantities, including crossing numbers, depend only on this combinatorial structure. Exhaustive enumeration of small order types can therefore support computational investigation of the rectilinear crossing number problem.

The unresolved problem asks for the smallest possible number of crossings among all straight-line drawings of the complete graph K_n obtained from n points in the plane. It is presented as an outstanding open question motivating the use of computational enumeration tools for small configurations.

References

As one example of an outstanding open question, we mention the rectilinear crossing number problem for the complete graph $K_n$: For a given set $S$ of $n$ points in the plane, draw all straight segments between pairs of points in~$S$, and count the pairs of segments that cross. What is the smallest number that can be obtained?

— NumPSLA -- An experimental research tool for pseudoline arrangements and order types  (2503.02336 - Rote, 4 Mar 2025) in Section 1, Introduction