Small mod-two area four-cycles in drawings with bounded pairwise intersections

Fix a bound on the number of meetings between any two edges of a complete graph drawn in the unit square. We prove that, as the number of vertices tends to infinity, some nonzero mod-two cycle has area tending to zero, uniformly over all such drawings. For at least four vertices the cycle can be chosen to be an abstract four-cycle, although its drawing may cross itself. The proof compares triangles through a common edge. Their parity regions have bounded shattering dimension, because a union of a fixed number of these triangles has only quadratically many complementary faces. An elementary sampling argument then finds two regions with small symmetric difference. We give an explicit, nonoptimal bound.