Sperner Theory

Konrad Engel

doi:10.1017/CBO9780511574719

The starting point of this book is Sperner's theorem, which answers the question: What is the maximum possible size of a family of pairwise (with respect to inclusion) subsets of a finite set? This theorem stimulated the development of a fast growing theory dealing with external problems on finite sets and, more generally, on finite partially ordered sets. This book presents Sperner theory from a unified point of view, bringing combinatorial techniques together with methods from programming, linear algebra, Lie-algebra representations and eigenvalue methods, probability theory, and enumerative combinatorics. Researchers and graduate students in discrete mathematics, optimisation, algebra, probability theory, number theory, and geometry will find many powerful new methods arising from Sperner theory.

‘The presentation is very good and the book should be understandable for a non-specialist.’

Source: European Mathematical Society

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Sperner Theory

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-x

1 - Introduction
pp 1-15

2 - Extremal problems for finite sets
pp 16-83

3 - Profile-polytopes for set families
pp 84-115

4 - The flow-theoretic approach in Sperner theory
pp 116-178

5 - Matchings, symmetric chain orders, and the partition lattice
pp 179-207

6 - Algebraic methods in Sperner theory
pp 208-303

7 - Limit theorems and asymptotic estimates
pp 304-331

8 - Macaulay posets
pp 332-389

Notation
pp 390-394

Bibliography
pp 395-412

Index
pp 413-417

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