Abstract
AbstractHill's Conjecture states that the crossing number of the complete graph in the plane (equivalently, the sphere) is . Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely , thus matching asymptotically the conjectured value of . Let denote the crossing number of a graph in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of is . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if . We construct drawings of in the projective plane that disprove this.
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