  Let $n$, $q$ and $r$ be positive integers, and let $K_N^n$ be the
$n$-skeleton of an $(N-1)$-simplex. We show that for $N$ sufficiently large
every embedding of $K_N^n$ in $\mathbb{R}^{2n+1}$ contains a link
$L_1\cup\cdots\cup L_r$ consisting of $r$ disjoint $n$-spheres, such that the
linking number $link(L_i,L_j)$ is a nonzero multiple of $q$ for all $i\neq j$.
This result is new in the classical case $n=1$ (graphs embedded in
$\mathbb{R}^3$) as well as the higher dimensional cases $n\geq 2$; and since it
implies the existence of a link $L_1\cup\cdots\cup L_r$ such that
$|link(L_i,L_j)|\geq q$ for all $i\neq j$, it also extends a result of Flapan
et al. from $n=1$ to higher dimensions. Additionally, for $r=2$ we obtain an
improved upper bound on the number of vertices required to force a
two-component link $L_1\cup L_2$ such that $link(L_1,L_2)$ is a nonzero
multiple of $q$. Our new bound has growth $O(nq^2)$, in contrast to the
previous bound of growth $O(\sqrt{n}4^nq^{n+2})$.
