volume 29, issue 3, P395-408 2003
DOI: 10.1007/s00454-002-0755-6
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## Abstract: In 1980 Katchalski and Lewis showed the following: if each three members of a family of disjoint translates in the plane are met by a line, then there exists a line meeting all but at most k members of F, where k is some positive constant independent of the family. They also showed that k can be taken to be less than 603, and conjectured that k = 2 is a universal bound for all such families. In 1990 Tverberg improved the upper bound by showing that k ≤ 108 holds. We make further improvements on the upper boun…

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