A cube tiling of dimension eight with no facesharing
\textit{O.-H. Keller} [J. Reine Angew. Math. 163, 231-248 (1930; JFM 56.1120.01)] conjectured that any tiling of \(n\)-dimensional space by translates of the unit \(n\)-cube contains a pair of cubes that share a complete common \((n-1)\)-dimensional face. \textit{O. Perron} [Math. Z. 46, 1-26 (1940; Zbl 0022.20205), ibid. 161-180 (1940; Zbl 0022.30802)] proved Keller's conjecture correct in dimensions \(\leq 6\). Corrádi and Szabo introduced the graph \(G_n\) with \(4^n\) vertices: the \(n\)-tuples \({\mathbf X}=(x_1, x_2, \dots,x_n)\) whose entries are \(0,1,2\) or 3; two verices \({\mathbf X}\) and \({\mathbf Y}\) are joined by an edge when \(|x_i-y_i |=2\) for at least one \(i\). They showed that Keller's conjecture is true in dimension \(n\) if and only if there is no clique of size \(2^n\) in \(G_n\) in which every pair of vertices differ in at least two coordinates. The author proves that Keller's conjecture is false for \(n=8\). He finds a clique in \(G_8\) with 256 vertices; indeed he lists them!
scientific article; zbMATH DE number 1883412