Studies On Related Problems Of Sperner Families

Sperner theory is an important part of extreme combinatorics. Erdos and Sperner set it up in1920s. Many classic methods such as Lubell-Yamamoto-Meschalkin inequality, probability method and so on are applied. And there are a lot of applications in hypergraph theory and many other branches of mathematics.This paper contains five parts:In the first part of the thesis, we introduce the concept of Sperner and the importance of Sperner theory. Then, we achieve the extreme value of uncomplement Sperner family by two methods.In the second part of this thesis, we mainly study the properties of a series of pairwise in-comparable families. First, we discuss the bound of a series pairwise incomparable families. Then, we use the similar method to prove a series pairwise incomparable families with other conditions. We also give the relation of a series of pairwise incomparable complemented families and uncomplement families.In the third part of this thesis, we apply induction method to prove the bound of a series of pairwise uncomplement Sperner families. Then, we discuss the bound of a series of disjoint Sperner families.In the four part of this thesis, we focus on the extreme problem of two pairwise incomparable intersecting families. We apply partition method to construct new families. A bound of two incomparable intersecting families is obtained.In the last part, we give a summary of this paper.