INTERSECTION PROPERTIES OF BOXES IN Rd.

摘要

A family of sets is called n-pierceable if there exists a set of n points such that each member of the family contains at least one of the points. Helly's theorem on intersections of convex sets concerns families of 1-pierceable sets. This note gives a complete solution to the following Helly-type problem: If d and n are positive integers,what is the least h = h(d, n) such that a family of boxes in d-space is n-pierceable whenever each of its k-membered subfamilies is n-pierceable. (Author)