Long Symmetric Chains in the Boolean Lattice

Let 2[n]be the poset of all subsets of a set with n elements ordered by inclusion. A long chain in this poset is a chain ofn−1 subsets starting with a subset with one element and ending with a subset withn−1 elements. In this paper we prove: Given any collection of at mostn−2 skipless chains in 2[n], there exists at least one (but sometimes not more than one) long chain disjoint from the chains in the collection. Furthermore, fork⩾3, given a collection ofn−kskipless chains in 2[n], there are at leastkpairwise disjoint long chains which are also disjoint from the given chains.