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arXiv · 1910.03740

The Resolution of Keller's Conjecture

Joshua Brakensiek·Marijn Heule·John Mackey·David Narváez
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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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Joshua Brakensiek, Marijn Heule, John Mackey, David Narváez. 2019-10-09. The Resolution of Keller's Conjecture. https://arxiv.org/abs/1910.03740

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