Proof of the Katchalski-Lewis Transversal Conjecture for T(3)-Families of Congruent Discs

A family of disjoint closed congruent discs is said to have property T(3) if to every triple of discs there exists a common line transversal. Katchalski and Lewis [10] proved the existence of a constant mdisc such that to every family of disjoint closed congruent discs with property T(3) a straight line can be found meeting all but at most mdisc of the members of the family. They conjectured that this is true even with mdisc = 2. On one hand Bezdek [1] proved mdisc ≥ 2 in 1991 and on the other hand Kaiser [9] showed mdisc ≤ 12 in a recent paper. The present work is devoted to proving this conjecture showing that mdisc ≤ 2.