Line Transversals in Large T(3)- and T(4)-Families of Congruent Discs

Abstract

A family of closed discs is said to have property T(k) if to every subset of at most k discs there belongs a common line transversal. A family of discs is said to be d-disjoint, d≥1, if the mutual distance between the centers of the discs is larger than d. It is known that a d-disjoint T(3)-family ℱ of unit diameter discs has a line transversal if $d=\sqrt{2}$. Similarly, a d-disjoint T(4)-family has a line transversal if $d=2/\sqrt{3}$. Both results are sharp in d, i.e., they do not hold for smaller values of d. The main result of this paper is that while the above lower bounds on d cannot be relaxed in general, some reduction of d can be compensated by imposing a proper d-dependent lower bound on the size of the family in both cases.