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.

中文翻译：

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解决凯勒猜想

我们考虑三个图，$G_{7,3}$、$G_{7,4}$ 和 $G_{7,6}$，与第 7 维的凯勒猜想有关。如果和仅当至少一个图表包含大小为 $2^7 = 128$ 的集团时。我们提出了一种自动方法来解决这个猜想，通过将这样一个集团的存在编码为一个命题公式。我们应用可满足性求解结合对称破坏技术来确定不存在这样的集团。这个结果意味着 $\mathbb{R}^7$ 的每个单位立方体平铺都包含一个面共享的立方体对。由于 $\mathbb{R}^8$ 的无面共享单位立方体平铺存在（我们也验证了），这完全解决了凯勒的猜想。