Intrinsic linking with linking numbers of specified divisibility

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 $i\mathrm{\ne }j$. This result is new in the classical case $n=1$ (graphs embedded in ${\mathbb{R}}^3$) as well as the higher dimensional cases $n\ge 2$; and since it implies the existence of a link $L_1\cup \cdots \cup L_r$ such that $\mathrm{\mid }link(L_i,L_j)\mathrm{\mid }\ge q$ for all $i\mathrm{\ne }j$, 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 $O(n{q}^2)$, in contrast to the previous bound of growth $O(\sqrt{n}{4}^n{q}^{n+2})$.