New Bounds on the Katchalski—Lewis Transversal Problem

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 bound of k , showing that k≤ 22 . Finally, we give a construction of a family of disjoint translates of a parallelogram, each three being met by a line, but where any line misses at least four members. This provides a counterexample to the Katchalski—Lewis conjecture.