Nested Chain Partitions of Hamiltonian Filters

LetPbe a poset, consisting of all setsX⊆[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}, thenPis a nested chain order. We examine the Sperner-type properties of such posets and provide further support for a conjecture of Lih.