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It and are two families of pairwise disjoint simple closed curves in the plane such that each curve in intersects each curve in, then the total number of points of intersection in is at least 2(m−1)n, where, and this bound is best possible. We use this to show that the cartesian product of two 5-cycles has crossing number 15.

Intersections of curve systems and the crossing number ofC 5 ×C 5