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

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.Comment: 26 pages, 4 figures. v3: references added, some typos corrected, order of Thms 1.4 and 1.5 reversed, other minor changes in response to referee's comments. v2: added reference to arXiv:0705.2026 and updated abstract and introduction in view of that paper; improved bound in Thm 1.4 from O(p^4) to O(p^2); some additional discussion of results; typos correcte