Journal of Combinatorial Theory Series A ( IF 1.1 ) Pub Date : 2020-02-27 , DOI: 10.1016/j.jcta.2020.105236 Oswin Aichholzer , Martin Balko , Thomas Hackl , Jan Kynčl , Irene Parada , Manfred Scheucher , Pavel Valtr , Birgit Vogtenhuber
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 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 have been of order and , respectively. We show that , obtaining the first superlinear lower bound on .
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.
中文翻译:
5孔数量的超线性下界
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 已经有秩序 和 , 分别。我们证明,获得第一个超线性下界 。
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。