Summary: 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 J. Eckhoff [Discrete Comput. Geom. 9, No. 2, 203–214 (1993; Zbl 0772.52002)], 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. Three sets \(A,B,C\) form a tight triple if \(\mathrm{conv}(A\cup B)\cap\mathrm{conv}(A\cup C)\cap\mathrm{conv}(B\cap C)\neq\emptyset\). This notion was first introduced by Holmsen, who 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 \(1\leq j\leq 6\) 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.