# Improvement on line transversals of families of connected sets in the plane

## TL;DR

This paper proves a new geometric result about families of connected sets in the plane, showing the existence of three concurrent lines intersecting all sets under certain conditions, extending previous work with fewer families and including unbounded sets.

## Contribution

It introduces a more general, colorful version of the line transversals problem, reducing the number of families from six to five and allowing parallel lines for unbounded sets, improving prior results.

## Key findings

- Existence of three concurrent lines intersecting all sets in the family.
- Extension to unbounded families with parallel lines.
- Reduction from six to five families in the colorful version.

## Abstract

Three lines are concurrent if they intersect at a single point. In this paper I prove that if $F$ is a bounded family of compact connected sets in the plane, such that every three sets in $F$ can be pierced by a single line, then there exists three concurrent lines in the plane such that the union of the three lines intersect every member of $F$. This had previously only been proven for lines that are not required to be concurrent by McGinnis and Zerbib in arXiv:2103.05565v2. In fact, I prove a more general, ``colorful'' version of this result: If $F_1, \dots , F_5$ are bounded families of compact connected sets in the plane such that every three sets, chosen from three distinct families $F_i$, can be pierced by a single line, then there exists $1 \leq j \leq 5$ and three concurrent lines, such that the union of the three lines intersect every member of $F_j$. McGinnis and Zerbib had 6 families instead of 5, so I also improve their result in this respect. Moreover, the result can also be extended to unbounded families, if we allow the piercing lines to be parallel.