Extremal Problems Of Intersecting And Sperner Families Of Sperner Theory

The research of extremal set theory is devoted to combinatorical problems of finite sets, and it is one of the most important branches in Combinatorics. Extremal set theory can be traced back to the famous Sperner theory which was given by the well known mathematician Sperner in the year 1928. From then on, many mathematicians such as Dilworth, Katona, Kleitman etc spent much time on this kind of problems and gave many classical results. The E-K-R theorem on intersecting Sperner families which was given by Erdos, Ko and Rado in 1961 is one of the most important results. Later on, there are a lot of references on the anolog, generalization and application of the E-K-R theorem. In 1968, Milner gave a classical result by drawing a prameter into intersecting Sperner families ; in 1980, Lih gave Sperner theorem over a subset.In this thesis we combine the results of Milner and Lih, led into two parameters and investigated the general intersecting Sperner families over a subset and several corollaries related to the problem as well. Besides, we apply the above results to complemented Sperner families over a subset.The first section of this thesis simply introduces some conceptions of extremal set theory and states the research meaning and developments of Sperner theory.In the second section, we mainly introduce all kinds of extremal problems of intersecting Sperner families.In the last section we state some typical extremal problems in hypergraph language.