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 m_disc 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 m_disc of the members of the family. They conjectured that this is true even with m_disc = 2. On one hand Bezdek 1] proved m_disc ≥ 2 in 1991 and on the other hand Kaiser 9] showed m_disc ≤ 12 in a recent paper. The present work is devoted to proving this conjecture showing that m_disc ≤ 2.