Keller's Conjecture

Keller conjectured that tiling an -dimensional space with -dimensional hypercubes of equal size yields an arrangement in which at least two hypercubes have an entire -dimensional "side" in common. This conjecture generalizes Minkowski's conjecture.

Corrádi and Szabó (1990) reformulated the conjecture by showing that if there exists a clique of size (the largest possible) in the class of graphs which have now become known as Keller graphs, then Keller's conjecture is false in that dimension. However, note that the nonexistence of such a clique does not necessarily imply the truth of the conjecture, only that no counterexample exists with hypercubes whose coordinates are integers or half-integers (Debroni et al. 2011).

Perron (1940) proved Keller's conjecture to be true in dimensions six and less, and it has been shown to be false in dimensions 8, 10, and 12 by Lagarias and Shor (1992), who found a cliques of size and in the 10-and 12-dimensional Keller graphs, respectively, and by Mackey (2002), who found a clique of size in the Keller graph of dimension eight. Debroni et al. (2011) recently showed the clique number of the 7-dimensional Keller graph is 124, thus suggesting (but not establishing) that Keller's conjecture is false in that dimension.