The Resolution of Keller's Conjecture

Joshua Brakensiek, Marijn Heule, John Mackey

Published 2019-10-09Version 1

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 $\mathbb{R}^7$ contains a facesharing pair of cubes. Since there is a faceshare-free unit cube tiling of $\mathbb{R}^8$, this completely resolves Keller's conjecture.