Intrinsic linking with linking numbers of specified divisibility

Let n , q and r be positive integers, and let K N n be the n -skeleton of an ( N  − 1) -simplex. We show that for N sufficiently large every embedding of K N n in ℝ 2 n  + 1 contains a link consisting of r disjoint n -spheres, such that every pairwise linking number is a nonzero multiple of q . This result is new in the classical case n  = 1 (graphs embedded in ℝ 3 ) as well as the higher dimensional cases n  ≥ 2 ; and since it implies the existence of an r -component link with all pairwise linking numbers at least q in absolute value, it also extends a result of Flapan et al. from n  = 1 to higher dimensions. Additionally, for r  = 2 we obtain an improved upper bound on the number of vertices required to force a two-component link with linking number a nonzero multiple of q . Our new bound has growth O ( n q 2 ) , in contrast to the previous bound of growth O (√( n )4 n q n + 2 ) .