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Creators/Authors contains: "Mackey, John"

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  1. ; ; (, Graphs and Combinatorics)
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  2. ; ; ; (, Journal of Automated Reasoning)
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  3. ; ; ; (, International Joint Conference on Automated Reasoning IJCAR 2020)
    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 ℝ^7 contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of ℝ^8 exists (which we also verify), this completely resolves Keller’s conjecture. 
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