AZ-identities and Strict 2-part Sperner Properties of Product Posets

TL;DR

This paper extends AZ identities to regular posets and characterizes maximum 2-part Sperner systems in product posets, advancing extremal set theory and combinatorial inequalities.

Contribution

It introduces AZ-type identities for regular posets and characterizes maximum 2-part Sperner systems in a broad class of product posets.

Findings

AZ identities derived for regular posets

Characterization of maximum 2-part Sperner systems

Extension of Sperner's theorem to product posets

Abstract

One of the central issues in extremal set theory is Sperner's theorem and its generalizations. Among such generalizations is the best-known BLYM inequality and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM inequality into an identity. Sperner's theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.