Covering Problem of Rado

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The covering problem of Rado is an unsolved problem in geometry concerning covering planar sets by squares. It was formulated in 1928 by Tibor Radó and has been generalized to more general shapes and higher dimensions by Richard Rado. In a letter to Wac¿aw Sierpi¿ski, motivated by some results of Giuseppe Vitali, Tibor Radó observed that for every covering of a unit interval, one can select a subcovering consisting of pairwise disjoint intervals with total length at least 1/2 and that this number cannot be improved. He then asked for an analogous statement in the plane.Radó proved that this number is at least 1/9 and conjectured that it is at least 1/4 a constant which cannot be further improved. This assertion was proved for the case of equal squares independently by A. Sokolin, R. Rado, and V. A. Zalgaller. However, in 1973, Miklós Ajtai disproved Radó's conjecture, by constructing a system of squares of two different sizes for which any subsystem consisting of disjoint squares covers the area at most 1/4 ¿ 1/1728 of the total area covered by the system.