Let P be a finite set of points in the plane in general position, that is, no three points of P are on a common line. We say that a set H of five points from P is a 5-hole in P if H is the vertex set of a convex 5-gon containing no other points of P. For a positive integer n, let $h_5(n)$ be the minimum number of 5-holes among all sets of n points in the plane in general position.

Despite many efforts in the last 30 years, the best known asymptotic lower and upper bounds for $h_5(n)$ have been of order $\mathrm{\Omega }(n)$ and $O(n^2)$, respectively. We show that $h_5(n)=\mathrm{\Omega }(n{\log }^{4/5}n)$, obtaining the first superlinear lower bound on $h_5(n)$.

The following structural result, which might be of independent interest, is a crucial step in the proof of this lower bound. If a finite set P of points in the plane in general position is partitioned by a line ℓ into two subsets, each of size at least 5 and not in convex position, then ℓ intersects the convex hull of some 5-hole in P. The proof of this result is computer-assisted.

令P为平面中一般位置上的有限点集，即P上没有三个点在同一条线上。我们说，如果H是一个不包含P的其他点的凸5边形的顶点集，那么从P出发的五个点的集合H就是P中的5孔。对于正整数n，令$H_5（ñ）$是一般位置平面中所有n个点集中的最小5孔数。

尽管在过去30年中进行了许多努力，但最著名的渐近下界和上界 $H_5（ñ）$ 已经有秩序 $\mathrm{\Omega }（ñ）$ 和 $Ø（{ñ}^2）$， 分别。我们证明$H_5（ñ）=\mathrm{\Omega }（ñ{\mathrm{日志}}^{4/5}ñ）$，获得第一个超线性下界 $H_5（ñ）$。

以下结构性结果可能是具有独立利益的，是证明这一下界的关键步骤。如果一个有限集合P在一般位置平面上的点的通过线划分ℓ成两个子集，每个尺寸的至少5，而不是在凸起位置，然后ℓ相交在一些5-孔的凸包P。此结果的证明是计算机辅助的。