Arcs on punctured disks intersecting at most twice with endpoints on the boundary

Abstract

Let $D_n$ be the $n$-punctured disk. We prove that a family of essential simple arcs starting and ending at the boundary and pairwise intersecting at most twice is of size at most $\left(\genfrac{}{}{0ex}{}{n+1}{3}\right)$. On the way, we also show that any nontrivial square complex homeomorphic to a disk whose hyperplanes are simple arcs intersecting at most twice must have a corner or a spur.