Holes or Empty Pseudo-Triangles in Planar Point Sets

TL;DR

This paper determines exact and bounded values for the minimum number of points needed to guarantee the existence of certain empty convex polygons or pseudo-triangles in planar point sets, extending previous geometric results.

Contribution

It provides exact values for E(k, 5) and E(5, 5), bounds for E(k, 6) and E(6, 5), and extends results on F(k, 5) and F(k, 6) in planar geometry.

Findings

Exact values of E(k, 5) and E(5, 5) are determined.

Bounds on E(k, 6) and E(6, 5) are established.

Exact values of F(k, 5) and F(k, 6) are obtained, with bounds on F(k, 7).

Abstract

Let $E(k, \ell)$ denote the smallest integer such that any set of at least $E(k, \ell)$ points in the plane, no three on a line, contains either an empty convex polygon with $k$ vertices or an empty pseudo-triangle with $\ell$ vertices. The existence of $E(k, \ell)$ for positive integers $k, \ell\geq 3$, is the consequence of a result proved by Valtr [Discrete and Computational Geometry, Vol. 37, 565--576, 2007]. In this paper, following a series of new results about the existence of empty pseudo-triangles in point sets with triangular convex hulls, we determine the exact values of $E(k, 5)$ and $E(5, \ell)$, and prove bounds on $E(k, 6)$ and $E(6, \ell)$, for $k, \ell\geq 3$. By dropping the emptiness condition, we define another related quantity $F(k, \ell)$, which is the smallest integer such that any set of at least $F(k, \ell)$ points in the plane, no three on a line, contains a convex polygon with $k$ vertices or a pseudo-triangle with $\ell$ vertices. Extending a result of Bisztriczky and T\'oth [Discrete Geometry, Marcel Dekker, 49--58, 2003], we obtain the exact values of $F(k, 5)$ and $F(k, 6)$, and obtain non-trivial bounds on $F(k, 7)$.