Intrinsic linking with linking numbers of specified divisibility - Tuffley, Christopher (Ars mathematica contemporanea, 2019)
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 ?$\mathbb{R}^{2n + 1}$? 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 ?$\mathbb{R}^3$?) 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 ?$O(nq^2)$?, in contrast to the previous bound of growth ?$O(\sqrt{n}4^nq^{n + 2})$?.
Keywords: complete n-complex; intrinsic linking; intrinzično povezovanje; polni n-kompleks; Ramsey theory; Ramseyeva teorija