Define the complete n-complex on N vertices, K n N , 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 λ 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, λ or q.