Cutting a Set of Disks by a Line with Leaving Many Intact Disks in Both Sides

Cut by a line the union of given disjoint disks in the plane so that both sides of the line contain many intact disks. At least how many intact disks can we leave in either side? It is proved that there is a family of infinitely many disjoint disks in the plane for which every line has a side that contains at most one intact disk. On the other hand, for any family of n disjoint disks, there is a circle C such that both the interior and the exterior of C contain n/4−o(n) intact disks.