The Sperner property for posets: A probabilistic approach

AbstractMotivated by the problem of estimating the age (in generations) of a population that evolves according to the Galton-Watson process, we consider graded partially ordered sets on which a probability measure is defined. By looking at the antichain of maximal probability, one derives a new proof of Sperner's lemma (1928) on the subsets of a set. More importantly, the technique of proof lends itself to generalizations to infinite posets and provides sufficient conditions on the probability measure and the order relation so that the poset has the Sperner property and/or the rank unimodality. These results are extended to k-families and the strong Sperner property and related to some work by Erdös (1945), Dilworth (1950), Baker (1969), Kleitman, Edelberg and Lubell (1971), Greene and Kleitman (1976), Stanley (1980), Griggs (1980), and Pouzet and Rosenberg (1981)