Nested chain partitions of Hamiltonian filters

Let P be a poset, consisting of all sets X subset of or equal to [n] = {1, 2, ..., n} which contain at least one of a given collection F of 2-subsets of [n], ordered by inclusion. By modifying a construction of Greene and Kleitman, we show that if F is hamiltonian, that is, contains {1, 2}, {2, 3}, ..., (n - 1, n) and {1, n}, then P is a nested chain order. We examine the Sperner-type properties of such posers and provide further support for a conjecture of Lih. (C) 1998 Academic Press, Inc.