Given a set of points $S\subseteq {\mathbb{R}}^2$, a subset $X\subseteq S$ with $|X|=k$ is called k-gon if all points of X lie on the boundary of the convex hull of X, and k-hole if, in addition, no point of $S\setminus X$ lies in the convex hull of X. We use computer assistance to show that every set of 17 points in general position admits two disjoint 5-holes, that is, holes with disjoint respective convex hulls. This answers a question of Hosono and Urabe (2001). We also provide new bounds for three and more pairwise disjoint holes.

In a recent article, Hosono and Urabe (2018) present new results on interior-disjoint holes – a variant, which also has been investigated in the last two decades. Using our program, we show that every set of 15 points contains two interior-disjoint 5-holes.

Moreover, our program can be used to verify that every set of 17 points contains a 6-gon within significantly smaller computation time than the original program by Szekeres and Peters (2006). Another independent verification of this result was done by Marić (2019).

给定一点 $\mathrm{小号}\subseteq {\mathbb{[R}}^2$，一个子集 $X\subseteq \mathrm{小号}$ 与 $|X|=ķ$被称为K-坤如果的所有点X谎言的凸包的边界上X，和K-孔如果，此外，没有点的$\mathrm{小号}\setminus X$位于X的凸包中。我们使用计算机辅助来表明，在一般位置上的每17个点集都允许有两个不相交的5孔，即具有不相交的凸包的孔。这回答了Hosono和Urabe（2001）的问题。我们还为三个或更多个成对的不相交孔提供了新的边界。

Hosono和Urabe（2018）在最近的一篇文章中介绍了内部不相交孔的新结果–一种变体，最近二十年来也进行了研究。使用我们的程序，我们表明每15个点集包含两个内部不相交的5孔。

此外，我们的程序可以用来验证每17个点集在比Szekeres和Peters（2006）的原始程序更短的计算时间内就包含一个6边形。Marić（2019）对此结果进行了另一项独立验证。