On the maximum number of tangencies among $1$-intersecting curves

Abstract

According to a conjecture of Pach, there are $O(n)$ tangent pairs among any family of $n$ Jordan arcs in which every pair of arcs has precisely one common point and no three arcs share a common point. This conjecture was proved for two special cases, however, for the general case the currently best upper bound is only $O(n^{7/4})$. This is also the best known bound on the number of tangencies in the relaxed case where every pair of arcs has \emph{at most} one common point. We improve the bounds for the latter and former cases to $O(n^{5/3})$ and $O(n^{3/2})$, respectively. We also consider a few other variants of these questions, for example, we show that if the arcs are \emph{$x$-monotone}, each pair intersects at most once and their left endpoints lie on a common vertical line, then the maximum number of tangencies is $\Theta(n^{4/3})$. Without this last condition the number of tangencies is $O(n^{4/3}(\log n)^{1/3})$, improving a previous bound of Pach and Sharir. Along the way we prove a graph-theoretic theorem which extends a result of Erd\H{o}s and Simonovits and may be of independent interest.