Intrinsically triple linked complete graphs (Q5943951)

The existence of so-called intrinsically linked graphs, these with the property that every embedding of \(G\) in \(\mathbb{R}^3\) contains a nontrivial link, is shown in [\textit{J. H. Conway}, \textit{C. McA. Gordon}, J. Graph Theory 7, 445-453 (1983; Zbl 0524.05028)] and [\textit{H. Sachs}, Colloq. Math. Soc. János Bolyai 37, 649-662 (1984; Zbl 0568.05026)]. The complete graphs \(K_6\) and \(K_7\) are shown to have this property. In this paper it is proved that \(K_{10}\) is intrinsically triple linked while \(K_9\) is not. An illustration of an embedding of \(K_9\) having no triple link is also given.