The $(2,2)$ and $(4,3)$ properties in families of fat sets in the plane

TL;DR

This paper establishes uniform bounds on the minimum number of points needed to intersect all sets in certain families of fat sets in the plane, extending classical results on disks with specific intersection properties.

Contribution

It introduces constant upper bounds on piercing numbers for families of r-fat sets satisfying the (2,2) or (4,3) properties, generalizing previous results on disks.

Findings

Bounded piercing numbers for r-fat sets with (2,2) property

Bounded piercing numbers for r-fat sets with (4,3) property

Extension of classical disk intersection results

Abstract

A family of sets satisfies the $(p,q)$ property if among every $p$ members of it some $q$ intersect. Given a number $0<r\le 1$, a set $S\subset \mathbb{R}^2$ is called $r$-fat if there exists a point $c\in S$ such that $B(c,r) \subseteq S\subseteq B(c,1)$, where $B(c,r)\subset \mathbb{R}^2$ is a disk of radius $r$ with center-point $c$. We prove constant upper bounds $C=C(r)$ on the piercing numbers in families of $r$-fat sets in $\mathbb{R}^2$ that satisfy the $(2,2)$ or the $(4,3)$ properties. This extends results by Danzer and Karasev on the piercing numbers in intersecting families of disks in the plane, as well as a result by Kyn\v{c}l and Tancer on the piercing numbers in families of units disks in the plane satisfying the $(4,3)$ property.