Intrinsically n-linked Complete Bipartite Graphs

We prove that every embedding of $K_{2n+1,2n+1}$ into ${\mathbb{R}}^3$ contains a non-split link of $n$-components. Further, given an embedding of $K_{2n+1,2n+1}$ in ${\mathbb{R}}^3$, every edge of $K_{2n+1,2n+1}$ is contained in a non-split $n$-component link in $K_{2n+1,2n+1}$.