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

Three lines are concurrent if they intersect at a single point. In this paper we 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 [8] . In fact, we prove a more general, “colorful” version of this result: If F 1 , … , 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 ≤ j ≤ 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 we 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.