On the Transversal Helly Numbers of Disjoint and Overlapping Disks

A family of disks is said to have the property T (k) if any k members of the family have a common line transversal. We call a family of unit diameter disks t-disjoint if the distances between the centers are greater than t. We consider for each natural number k ≥ 3, the infimum tk of the distances t for which any finite family of t-disjoint unit diameter disks with the property T (k) has a line transversal. We determine exact values of t3 and t4, and give general lower and upper bounds on the sequence tk, showing that tk = O(1/k) as k → ∞.