Some Ramsey-type results on intrinsic linking of $n$–complexes

Abstract

Define the complete $n$–complex on $N$ vertices, $K_N^n$, to be the $n$–skeleton of an $\left(N-1\right)$–simplex. We show that embeddings of sufficiently large complete $n$–complexes in ${ℝ}^{2n+1}$ necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in ${ℝ}^3$ (the case $n=1$) to higher dimensions. In particular, we prove the existence of links of the following types: $r$–component links, with the linking pattern of a chain, necklace or keyring; $2$–component links with linking number at least $\lambda$ in absolute value; and $2$–component links with linking number a nonzero multiple of a given integer $q$. For fixed $n$ the number of vertices required for each of our results grows at most polynomially with respect to the parameter $r$, $\lambda$ or $q$.