On Covering Problems of Rado

Abstract

T. Rado conjectured in 1928 that if ${\mathcal S}$ is a finite set of axis-parallel squares in the plane, then there exists an independent subset ${\mathcal I}\subseteq$ ${\mathcal S}$ of pairwise disjoint squares, such that ${\mathcal I}$ covers at least 1/4 of the area covered by ${\mathcal S}$. He also showed that the greedy algorithm (repeatedly choose the largest square disjoint from those previously selected) finds an independent set of area at least 1/9 of the area covered by ${\mathcal S}$. The analogous question for other shapes and many similar problems have been considered by R. Rado in his three papers (1949, 1951 and 1968) on this subject. After 45 years (in 1973), Ajtai came up with a surprising example disproving T. Rado's conjecture. We revisit Rado's problem and present improved upper and lower bounds for squares, disks, convex sets, centrally symmetric convex sets, and others, as well as algorithmic solutions to these variants of the problem.