Connectivity of distance graphs (Q1196753)

Let \(K\supseteq\mathbb{Q}\) be a \(\mathbb{Z}\)-module and \(G\) a graph with a \(K\)- space as vertex set \(V\). If the edges of \(G\) are preserved under translations in \(V\) and \(G\) has more than one connected component, then the author shows that \(G\) has infinitely many components.