An asymptotic formula for the maximum size of an h-family in products of partially ordered sets

AbstractAn h-family of a partially ordered set P is a subset of P such that no h + 1 elements of the h-family lie on any single chain. Let S1, S2,… be a sequence of partially ordered sets which are not antichains and have cardinality less than a given finite value. Let Pn be the direct product of S1,…, Sn. An asymptotic formula of the maximum size of an h-family in Pn is given, where h=o(n) and n → ∞