AZ-style identities of downsets and upsets

The Ahlswede–Zhang identity is a sharpening of the well-known LYM inequality. To establish the identity, Ahlswede and Zhang introduced an important set function called AZ function. In this paper, we present some identities which establish a relation between the AZ function and cardinalities of downsets or upsets. Some connections of these identities with the original AZ identity are considered. Furthermore, these identities yield formulas concerning the size of a star in a downset.

A generalization of Sperner's theorem

Some generalizations of Sperner's theorem and of the LYM inequality are given to the case when A1,… At are t families of subsets of {1,…,m} such that a set in one family does not properly contain a set in another.