  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.
