On dividing planar sets into five parts

In 1933, K. Borsuk proposed to partition sets of diameter 1 into parts of smaller diameter. Borsuk's problem is now one of the most popular in combinatorial geometry. In 1956, H. Lenz refined Borsuk's problem by asking a question of a minimum diameter of a part in a partition of a set into a given number of parts. In 2010, V.P. Filimonov replaced the question of a minimum diameter by that of a minimum distance which is absent among the points of each part. Filimonov showed that in partitioning a set into five parts one can always avoid the distance 1/√3=0.577... We succeed in showing that the same is true of the distance (square root of (2 minus the square. root of 3)) =0.517... To do this, we develop a new technique for studying infinite universal covering systems, which is also of interest.