Some Problems In Sperner Thoeory

Sperner Theory is one of research branches in combinatorics, whose reserch object is posets, whose main content is to investgate the extremal problems on posets. Its source came from a theorem of Sperner in 1928: in a subset lattice, a size maximal rank set forms a size maximal antichain. After about one century's development Sperner's theorem has become into a systematic theory.The first chapter is a simple survey on this theory, including the relative notations, basic terminology and the main methods used later.In the second chapter, we introduce the definition of the q-degree log-concavity of a sequence, and give a series of linear transformations which preserve the q-degree log-concavity of a sequence.In the third chapter, we first give the definition of q-direct product, and then deduce the q-direct product theorem from product theorem: if (Q, v) and (P, w) are both q-degree log-concave and have the normalized matching property, then each q-direct product of them is q-degree log-concave and has the normalized matching property. By this theorem we prove that some subposets of L(V) are q-degree log-concave and have the normalized matching property.In the fourth chapter, we construct the nested chain decompositions for four subposets of the Boolean lattice.In the last chapter, we consider the LYM property and the local EKR property of the partial permutation poset B(n,n).