Line transversals in families of connected sets the plane

TL;DR

This paper proves that for a family of connected sets in the plane where every three are intersected by a line, three lines suffice to intersect all sets, improving previous bounds and providing a colorful version of the result.

Contribution

It establishes a new bound that three lines can intersect all sets under certain conditions, advancing the understanding of line transversals in geometric families.

Findings

Proves that three lines intersect all sets in the family.

Improves the bound from 1/8 to 1/3 in Holmsen's result.

Provides a colorful version of the line transversal theorem.

Abstract

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that, under the same condition, there are four lines intersecting all the sets. In fact, we prove a colorful version of this result, under weakened conditions on the sets. A triple of sets $A,B,C$ in the plane is said to be a {\em tight} if $\textrm{conv}(A\cup B)\cap \textrm{conv}(A\cup C)\cap \textrm{conv}(B\cap C)\neq \emptyset.$ This notion was first introduced by Holmsen, where he showed that if $\mathcal{F}$ is a family of compact convex sets in the plane in which every three sets form a tight triple, then there is a line intersecting at least $\frac{1}{8}|\mathcal{F}|$ members of $\mathcal{F}$. Here we prove that if $\mathcal{F}_1,\dots,\mathcal{F}_6$ are families of compact connected sets in the plane such that every three sets, chosen from three distinct families $\mathcal{F}_i$, form a tight triple, then there exists $1\le j\le 6$ and three lines intersecting every member of $\mathcal{F}_j$. In particular, this improves $\frac{1}{8}$ to $\frac{1}{3}$ in Holmsen's result.