On Wegner’s inequality for axis-parallel rectangles

According to an old conjecture of Wegner, the piercing number of a set of axis-parallel rectangles in the plane is at most twice the independence number (or matching number) minus 1, that is, τ(F)≤2ν(F)−1. On the other hand, the current best upper bound, due to Corea et al. (2015), is a Ologlogν(F)2 factor away from the current best lower bound. From the other direction, lower bound constructions with τ(F)≥2ν(F)−4 are known. Here we exhibit families of rectangles with τ=7 and ν=4 and thereby show that Wegner’s inequality, if true, cannot be improved for ν=4. The analogous result for ν=3, due to Wegner, dates back to 1968. A key element in our proof is establishing a connection with the Maximum Empty Box problem: Given a set P of n points inside an axis-parallel box U in Rd, find a maximum-volume axis-parallel box that is contained in U but contains no points of P in its interior. Whereas our construction can be extended to any larger independence number (ν=5,6,…), its analysis remains open.