In this paper, we show that there is a Cantor set of initial conditions in the planar $4$‑body problem such that all four bodies escape to infinity in a finite time, avoiding collisions. This proves the Painlevé conjecture for the $4$‑body case, and thus settles the last open case of the conjecture.

中文翻译：

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平面四体问题中的非碰撞奇点

在本文中，我们证明了在平面$ 4 $体问题中存在Cantor初始条件集，以便所有四个体在有限的时间内逃逸到无穷大，从而避免碰撞。这证明了$ 4 $身体情况的Painlevé猜想，从而解决了该猜想的最后一个未决案例。