Sperner Theory / Konrad Engel.

Format:

Book

Author/Creator:

Engel, Konrad, author.

Series:

Encyclopedia of Mathematics and its Applications ; 65.

Encyclopedia of Mathematics and its Applications ; 65

Language:

English

Subjects (All):

Sperner theory.

Partially ordered sets.

Physical Description:

1 online resource (432 pages) : digital, PDF file(s).

Place of Publication:

Cambridge : Cambridge University Press, 1997.

Language Note:

English

Summary:

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.

Contents:

Cover; Half-title; Title; Copyright; Contents; Preface; 1 Introduction; 1.1 Sperner's theorem; 1.2 Notation and terminology; 1.3 The main examplesl; 2 Extremal problems for finite sets; 2.1 Counting in two different ways; 2.2 Partitions into symmetric chains; 2.3 Exchange operations and compression; 2.4 Generating families; 2.5 Linear independence; 2.6 Probabilistic methods; 3 Profile-polytopes for set families; 3.1 Full hereditary families and the antiblocking type; 3.2 Reduction to the circle; 3.3 Classes of families arising from Boolean expressions

4 The flow-theoretic approach in Sperner theory4.1 The Max-Flow Min-Cut Theorem and the Min-Cost Flow Algorithm; 4.2 The K-cutset problem; 4.3 The K-family problem and related problems; 4.4 The variance problem; 4.5 Normal posets and flow morphisms; 4.6 Product theorems; 5 Matchings, symmetric chain orders, and the partition lattice; 5.1 Definitions, main properties, and examples; 5.2 More part Sperner theorems and the Littlewood-Offord problem; 5.3 Coverings by intervals and sc-orders; 5.4 Semisymmetric chain orders and matchings; 6 Algebraic methods in Sperner theory

6.1 The full rank property and Jordan functions6.2 Peck posets and the commutation relation; 6.3 Results for modular, geometric, and distributive lattices; 6.4 The independence number of graphs and the Erdõs-Ko-Rado Theorem; 6.5 Further algebraic methods to prove intersection theorems; 7 Limit theorems and asymptotic estimates; 7.1 Central and local limit theorems; 7.2 Optimal representations and limit Sperner theorems; 7.3 An asymptotic Erdõs-Ko-Rado Theorem; 8 Macaulay posets; 8.1 Macaulay posets and shadow minimization; 8.2 Existence theorems for Macaulay posets

8.3 Optimization problems for Macaulay posets8.4 Some further numerical and existence results for chain products; 8.5 Sperner families satisfying additional conditions in chain products; Notation; Bibliography; Index

Notes:

Title from publisher's bibliographic system (viewed on 05 Feb 2026).

Includes bibliographical references (p. 395-412) and index.

ISBN:

1-139-88650-9

0-511-95964-8

1-107-10303-7

1-107-08864-X

0-511-57471-1

1-107-09483-6

1-107-09157-8

OCLC:

847979384