Line Transversals to Disjoint Disks

The main purpose of this paper is to prove the following result: Let $\mathcal F$ be a $\sqrt 2$-disjoint finite family of disks with the property that every four of them have a common line transversal. Then there exists a line transversal to all members of $\mathcal F$. Moreover, we prove that the number $\sqrt 2$ is sharp.