A new bound for the chromatic number of the rational five-space (Q2765092)

The chromatic number \(\chi \left( Q^{n}\right)\) of the rational \(n\)-space \(Q^{n}\) is defined as the chromatic number of the graph \(U_{n}=\left( V,E\right)\), with \(V=Q^{n}\) and \(\left( v,w\right) \in E\) iff \(d_{e}\left( u,v\right) =1,\) where \(d_{e}\) denotes the Euclidean distance. \textit{K. B. Chilakamarri} [Aequationes Math. 39, 146-148 (1990; Zbl 0705.05034)] showed that \(\chi \left( Q^{5}\right) \geq 6.\) In this paper a subgraph \(M\) of \(U_{n}\) with chromatic number 7 is presented, thus showing that \(\chi \left( Q^{5}\right) \geq 7.\) The graph \(M\) is presented by listing its vertices and edges and the proof that \(\chi \left( M\right) =7\) is by computer. A theoretical proof and a sketch of the graph can be found at \url{http://www.unit-distance-graphs.com}.