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horizontal and vertical segments intersected by curves

Source: Miklos Schweitzer 2020, Problem 4

December 1, 2020
geometrycombinatorics

Problem Statement

Consider horizontal and vertical segments in the plane that may intersect each other. Let nn denote their total number. Suppose that we have mm curves starting from the origin that are pairwise disjoint except for their endpoints. Assume that each curve intersects exactly two of the segments, a different pair for each curve. Prove that m=O(n)m=O(n).
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