Intersection properties of boxes in R d

A family of sets is calledn-pierceable if there exists a set ofn points such that each member of the family contains at least one of the points. Helly’s theorem on intersections of convex sets concerns 1-pierceable families. Here the following Helly-type problem is investigated: Ifd andn are positive integers, what is the leasth =h(d, n) such that a family of boxes (with parallel edges) ind-space isn-pierceable if each of itsh-membered subfamilies isn-pierceable? The somewhat unexpected solution is: (i)h(d, 2) equals3d for oddd and 3d?1 for evend; (ii)h(2, 3)=16; and (iii)h(d, n) is infinite for all (d, n) withd≧2 andn≧3 except for (d, n)=(2, 3).