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 less than or equal to 108 holds. We make further improvements on the upper bound of k, showing that k less than or equal to 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 KatchalskiL-Lewis conjecture.