Two Disjoint 5-Holes in Point Sets

TL;DR

This paper proves that any set of 17 points in general position always contains two disjoint 5-holes, using computer assistance, and provides new bounds for multiple disjoint holes, advancing combinatorial geometry knowledge.

Contribution

The authors use computational methods to confirm the existence of two disjoint 5-holes in 17-point sets, answering a longstanding open question and improving bounds for multiple disjoint holes.

Findings

Every set of 17 points admits two disjoint 5-holes.

Every set of 15 points contains two interior-disjoint 5-holes.

The program verifies 6-gons in 17 points faster than previous methods.

Abstract

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 Mari\'c (2019).