The Resolution of Keller's Conjecture [article]

Joshua Brakensiek, Marijn Heule, John Mackey, David Narváez
<span title="2020-12-02">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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
more &raquo; ... ube 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.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="">arXiv:1910.03740v4</a> <a target="_blank" rel="external noopener" href="">fatcat:en6u4c2cv5ee5oubinb54mupv4</a> </span>
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