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

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.