On the number of tangencies among 1-intersecting curves
arXiv:2305.138071 citationsh-index: 17
Originality Incremental advance
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Settles a conjecture for an important class of curves, providing a tight bound for tangency counts in combinatorial geometry.
The authors prove János Pach's conjecture that the number of touching pairs among 1-intersecting curves is linear in the number of curves, specifically for x-monotone curves.
Let $\cal C$ be a set of curves in the plane such that no three curves in $\cal C$ intersect at a single point and every pair of curves in $\cal C$ 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 $\cal C$ that touch each other is $O(|{\cal C}|)$. We prove this conjecture for $x$-monotone curves.