Volume 13, issue 3 (2013)

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

Christopher Tuffley

Algebraic & Geometric Topology 13 (2013) 1579–1612

Define the complete n–complex on N vertices, KNn, to be the n–skeleton of an (N 1)–simplex. We show that embeddings of sufficiently large complete n–complexes in 2n+1 necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in 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 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, λ or q.

intrinsic linking, $n$–complexes, Ramsey theory
Mathematical Subject Classification 2010
Primary: 57Q45
Secondary: 57M15, 57Q35
Received: 8 February 2012
Revised: 17 January 2013
Accepted: 24 January 2013
Published: 16 May 2013
Christopher Tuffley
Institute of Fundamental Sciences
Massey University
Private Bag 11222
Palmerston North 4442
New Zealand