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

TL;DR

This paper extends classical results on linking in large complete graphs to higher-dimensional complexes, showing that embeddings of sufficiently large n-complexes in R^{2n+1} necessarily contain complex links with specified properties.

Contribution

It proves that large complete n-complexes embedded in R^{2n+1} must contain links with various linking patterns and numbers, generalizing known 3D results to higher dimensions.

Findings

Existence of r-component links with chain, necklace, or keyring patterns.

Presence of 2-component links with large or specific linking numbers.

Number of vertices needed grows polynomially with parameters.

Abstract

Define the complete n-complex on N vertices to be the n-skeleton of an (N-1)-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 non-zero 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.