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partition 3d space into 3 subsets, mutual distance

Source: Austrian Polish 1987 APMC

April 30, 2020
distancepartitioncombinatorial geometrycombinatorics

Problem Statement

The Euclidian three-dimensional space has been partitioned into three nonempty sets A1,A2,A3A_1,A_2,A_3. Show that one of these sets contains, for each d>0d > 0, a pair of points at mutual distance dd.
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