Every Large Point Set contains Many Collinear Points or an Empty Pentagon

On empty pentagons and hexagons in planar point sets

We give improved lower bounds on the minimum number of $k$-holes (empty convex $k$-gons) in a set of $n$ points in general position in the plane, for $k=5,6$.

Empty pentagons in point sets with collinearities

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.

Disjoint Empty Convex Pentagons in Planar Point Sets

Harborth [{\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved that every set of 10 points in the plane, no three on a line, contains an empty convex pentagon. From this it follows that the number of disjoint empty convex pentagons in any set of $n$ points in the plane is least $\lfloor\frac{n}{10}\rfloor$. In this paper we prove that every set of 19 points in the plane, no three on a line, contains two disjoint empty convex pentagons. We also show that any set of $2m+9$ points in the plane, where $m$ is a positive integer, can be subdivided into three disjoint convex regions, tw…

Generalized pentagonal geometries

A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every line is incident with $k$ points, every point is incident with $r$ lines, and for each point $x$, there is a line incident with precisely those points that are not collinear with $x$. Here we generalize the concept by allowing the points not collinear with $x$ to form the point set of a Steiner system $S(2,k,w)$ whose blocks are lines of the geometry.

无需计算的五边形最小化

Pentagon Minimization without Computation

本文研究无三点共线平面点集凸五边形最小化问题，在无需计算的情况下，利用“平面点方程”并从统计视角结合，证明了$c_5$约为0.04508的新下界。

Happy endings for flip graphs

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.