Cube tiling and covering a complete graph

On the structure of cube tilings ofR3andR4

“…Finally, Keller's cube tiling conjecture turned out to be false. Using the results obtained by Corrádi and Szabó on cube tilings [1,2], Lagarias and Shor invalidated Keller's conjecture for dimensions d 10 [13], and next Mackey gave a counterexample to this conjecture in the dimension d = 8 [17]. (For d = 7 it is still open.)…”

“…∈ Z, which allowed us to conclude that C i = R. We give only an equality C i = R which, in each case considered below, will follow from the fact that there are [0, 1) …”

“…We say that a family of cubes [0, 1) d + S, S ⊂ R d , satisfies Keller's condition if for every two cubes [0, 1) d + t and [0, 1) …”

“…The most well-known structural theorem on cube tilings of R d is the Minkowski-Hajós theorem on the existence of a column of cubes C = {[0, 1) d + t + z: z ∈ Z}, t ∈ T , in a tiling [0, 1) …”

“…

A family of translates of the unit cube [0,1) is called a cube tiling of R d if cubes from this family are pairwise disjoint andA cube tiling of R d is blockable if there is a finite family of disjoint blocks B, |B| > 1, with the property that every cube from the tiling is contained in exactly one block of the family B. We construct a cube tiling T of R 4 which, in contrast to cube tilings of R 3 , is not blockable.

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