Abstract
We consider three graphs, G_{7,3}, G_{7,4}, and G_{7,6}, related to Keller’s conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size 2^7 = 128. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of mathbb {R}^7 contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of mathbb {R}^8 exists (which we also verify), this completely resolves Keller’s conjecture.
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