Intrinsically triple-linked graphs in RP³

Flapan--Naimi--Pommersheim showed that every spatial embedding of $K_{10}$, the complete graph on ten vertices, contains a non-split three-component link; that is, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}^3$. The work of Bowlin--Foisy and Flapan--Foisy--Naimi--Pommersheim extended the list of known intrinsically triple-linked graphs in $\mathbb{R}^3$ to include several other families of graphs. In this paper, we will show that while some of these graphs can be embedded 3-linklessly in $\mathbb{R}P^3$, $K_{10}$ is intrinsically triple-linked in $\mathbb{R}P^3$.