The Resolution of Keller's Conjecture
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We consider two graphs, G_7,3 and G_7,4, related to Keller's conjecture in dimension 7. We show, with computer assistance, that every maximal clique in either graph contains a facesharing pair of vertices. Doing so shows that every unit cube tiling of R^7 contains a facesharing pair of cubes. Since there is a faceshare-free unit cube tiling of R^8, this completely resolves Keller's conjecture.
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