Line transversals in large \(T(3)\)- and \(T(4)\)-families of congruent discs
Throughout \({\mathcal F}\) denotes a family of unit diameter discs in the (Euclidean) plane and the question is whether or not \({\mathcal F}\) has a common transversal (a line which intersects each member of \({\mathcal F}\)). \({\mathcal F}\) is \(d\)-disjoint if the distance between each pair of centres is (strictly) greater than \(d\) (\(d \geq 1\)). \({\mathcal F}\) is a \(T(k)\) family if every subset of at most \(k\) members has a common transversal. The following are known: Each \(1\)-disjoint \(T(5)\) family, each \(\sqrt{2}\)-disjoint \(T(4)\) family, and each \(2/\sqrt{3}\)-disjoint \(T(3)\) family has a common transversal. See [\textit{L. Danzer}, Arch. Math. 8, 347--351 (1957; Zbl 0078.35902) and \textit{K. Bezdek, T. Bisztriczky, B. Csikós} and \textit{A. Heppes}, Arch. Math. 87, No.~1, 86--96 (2006; Zbl 1105.52003)]. The author shows that \(d\) may be reduced in these theorems if the diameter of the family is made large: For any \(d > 2/\sqrt{3}\) (resp. \(d > 1\)) there exist a number \(D_3(d)\) (resp. \(D_4(d)\)) such that if diam\(({\mathcal F}) > D_3(d)\) (resp. diam\(({\mathcal F}) >D_4(d)\)) then each \(d\)-disjoint \(T(3)\) (resp. \(T(4)\)) family has a common transversal. It is also shown that the numbers \(2/\sqrt{3}\) and \(1\) are best possible.
scientific article; zbMATH DE number 5377154
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