New bounds on the Katchalski-Lewis transversal problem (Q1404508)

Let \({\mathcal F}\) be a family of disjoint translates of a compact convex set in the plane with the property that each three members of \({\mathcal F}\) are met by a line. Katchalski and Lewis showed in 1982 that there is a number \(k\), independent of \({\mathcal F}\), so that some line meets all but at most \(k\) members of \({\mathcal F}\). They also conjectured that one can choose \(k=2\). The present author sets a new record by showing that \(k\leq 22\), and he disproves the Katchalski-Lewis conjecture by constructing a family of disjoint translates of a parallelogram, each three met by a line, but where any line misses at least four members.