Discrete Helly-type theorems for pseudohalfplanes

TL;DR

This paper establishes new discrete Helly-type theorems for pseudohalfplanes, extending classical results to a broader class of geometric objects using combinatorial methods.

Contribution

It introduces novel Helly-type theorems for pseudohalfplanes and analyzes their hypergraph properties, expanding the understanding of geometric intersection patterns.

Findings

If every triple of pseudohalfplanes intersects at a point in P, then two points hit all pseudohalfplanes.

If every triple of points in P is contained in some pseudohalfplane, then two pseudohalfplanes cover all points.

The maximal chromatic number of the hypergraph families is determined.

Abstract

We prove discrete Helly-type theorems for pseudohalfplanes, which extend recent results of Jensen, Joshi and Ray about halfplanes. Among others we show that given a family of pseudohalfplanes $\cal H$ and a set of points $P$, if every triple of pseudohalfplanes has a common point in $P$ then there exists a set of at most two points that hits every pseudohalfplane of $\cal H$. We also prove that if every triple of points of $P$ is contained in a pseudohalfplane of $\cal H$ then there are two pseudohalfplanes of $\cal H$ that cover all points of $P$. To prove our results we regard pseudohalfplane hypergraphs, define their extremal vertices and show that these behave in many ways as points on the boundary of the convex hull of a set of points. Our methods are purely combinatorial. In addition we determine the maximal possible chromatic number of the regarded hypergraph families.