The Resolution of Keller's Conjecture

2019/10/09 by Joshua Brakensiek, Brakensiek, Joshua, Marijn Heule +5 · 1 voice · 3 citations
Computer Science · Mathematics · #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO) #Metric Geometry (math.MG) #cs.DM #cs.LO #math.CO #math.MG

paper · pdf · doi:10.48550/arxiv.1910.03740

arxiv published 2019/10/09 · arxiv updated 2023/04/17

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 2⁷ = 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 ℝ⁷ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of ℝ⁸ exists (which we also verify), this completely resolves Keller's conjecture.

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