Proof of the Katchalski-Lewis transversal conjecture for \(T(3)\)-families of congruent discs (Q2464362)

A family \({\mathcal F}\) of disjoint translates of an arbitrary planar compact convex set is said to have the \(T(k)\) (respectively, \(T\)) \textit{property} if every \(k\in {\mathbb N}\) members (respectively, all members) of this family are intersected by a common line. In [Proc. Am. Math. Soc. 79, 457--461 (1980; Zbl 0435.52006)] \textit{M. Katchalski} and \textit{T. Lewis} showed that there is a universal constant \(m\) (independent of \({\mathcal F}\)) such that if a family \({\mathcal F}\) has the \(T(3)\) property then there is a subfamily \({\mathcal G}\subset {\mathcal F}\), excluding at most \(m\) members of \({\mathcal F}\), which has the \(T\) property. They also got an upper bound for \(m\) and conjectured that \(m=2\). Families of special sets have been also considered. In this paper, the author considers the family of disjoint closed congruent discs in the Euclidean plane, for which the upper bound \(m_{\text{disc}}\leq 12\) and the lower bound \(m_{\text{disc}}\geq 2\) were obtained in different works by several authors. The goal of this paper is to show that, in this particular case of the disc, Katchalski-Lewis conjecture is true, i.e., \(m_{\text{disc}}=2\).