On the transversal Helly numbers of disjoint and overlapping disks

摘要

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 t k 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 t 3 and t 4, and give general lower and upper bounds on the sequence t k , showing that t k = O(1/k) as k → ∞.