Geodesic
Realization of all distances in a decomposition of the space $R^n$ into $n+1$ parts
Matematičeskie zametki, Tome 7 (1970) no. 3, pp. 319-323.

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Let the sets $A_1, A_2, \dots, A_{n+1}$ form a covering of the $n$-dimensional euclidean space $R^n$ ($n>1$). Then among these sets can be found a set $A_i$ containing, for every $d>0$, a pair of points such that the distance between them is equal to $d$. 
@article{MZM_1970_7_3_a8,
     author = {D. E. Raiskii},
     title = {Realization of all distances in a decomposition of the space $R^n$ into $n+1$ parts},
     journal = {Matemati\v{c}eskie zametki},
     pages = {319--323},
     publisher = {mathdoc},
     volume = {7},
     number = {3},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_7_3_a8/}
}
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D. E. Raiskii. Realization of all distances in a decomposition of the space $R^n$ into $n+1$ parts. Matematičeskie zametki, Tome 7 (1970) no. 3, pp. 319-323. http://geodesic.mathdoc.fr/item/MZM_1970_7_3_a8/