Line transversals in large T(3)- and T(4)-Families of congruent discs
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 = root 2. Similarly, a d-disjoint T(4)-family has a line transversal if d = 2/root 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.
line transversal disc d-disjoint