Independent collections of translates of boxes and a conjecture due to Grünbaum

A collection ofn setsA1, ...,An is said to beindependent provided every set of the formX1 ⋂ ... ⋂Xn is nonempty, where eachXi is eitherAi orAic. We give a simple characterization for when translates of a given box form an independent set inRd. We use this to show that the largest number of such boxes forming an independent set inRd is given by ⌊3d/2⌋ ford≥2. This settles in the negative a conjecture of Grünbaum (1975), which states that the maximum size of an independent collection of sets homothetic to a fixed convex setC inRd isd+1. It also shows that the bound of 2d of Rényiet al. (1951) for the maximum number of boxes (not necessarily translates of a given one) with sides parallel to the coordinate axes in an independent collection inRd can be improved for these special collections.