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 $\binom{n+1}{3}$. 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.

中文翻译：

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穿孔磁盘上的弧最多与边界上的端点相交

令$ D_n $为穿孔的$ n $。我们证明，基本的简单弧族在边界处开始和结束，并且成对相交最多两次，其大小最大为$ \ binom {n + 1} {3} $。顺便说一句，我们还表明，对于超平面为最多相交两次的简单圆弧的磁盘，任何非平凡的平方复同胚同形都必须具有拐角或正齿。