# The Sperner Property

• Richard P. Stanley
Chapter
Part of the Undergraduate Texts in Mathematics book series (UTM)

## Abstract

In this chapter we consider a surprising application of certain adjacency matrices to some problems in extremal set theory. An important role will also be played by finite groups in Chap.5, which is a continuation of the present chapter. In general, extremal set theory is concerned with finding (or estimating) the most or least number of sets satisfying given set-theoretic or combinatorial conditions. For example, a typical easy problem in extremal set theory is the following: what is the most number of subsets of an n-element set with the property that any two of them intersect? (Can you solve this problem?) The problems to be considered here are most conveniently formulated in terms of partially ordered sets or posets for short. Thus we begin with discussing some basic notions concerning posets.

## Keywords

Sperner Property Surprising Application Graded Poset Symmetric Chain Decomposition Maximal Chain
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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