On the number of tangencies among 1-intersecting curves

05/23/2023

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by   Eyal Ackerman, et al.

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Let C be a set of curves in the plane such that no three curves in C intersect at a single point and every pair of curves in C intersect at exactly one point which is either a crossing or a touching point. According to a conjecture of János Pach the number of pairs of curves in C that touch each other is O(| C|). We prove this conjecture for x-monotone curves.