Combinatorics and Partially Ordered Sets

Primarily intended for research mathematicians and computer scientists, this work also serves as a useful text for advanced students in either field. The book's most important contribution is to collect, organize, and explain the many theorems on partially ordered sets in a way that makes them available to the widest possible audience. (Mathematics)

William T. Trotter is Regents' Professor of Mathematics at Arizona State University and director of Combinatorics and Operations Research for Bell Communications Research in Morristown, New Jersey.

Preface

xi

Introduction to Dimension

Overview

1

(1)

Basic Notation for Partially Ordered Sets

2

(5)

Dilworth's Chain Covering Theorem

7

(2)

Extensions and Realizers

9

(3)

Standard Examples

12

(2)

Alternating Cycles

14

(4)

Dimension and Width

18

(3)

Interpolation and Lifting

21

(1)

Removal Theorems and Continuity

21

(2)

Lexicographic Sums and Hiraguchi's Inequality

23

(3)

Complements of Antichains

26

(3)

Reversing Critical Pairs

29

(4)

Crowns, Splits, Stacks, Sums, and Products

Overview

33

(1)

Generalized Crowns

34

(4)

Cartesian Products

38

(8)

Bipartite Posets

46

(2)

Splits and Quasi-Realizers

48

(3)

Stacks and Splits

51

(2)

Kelly's Dimension Products

53

(1)

Exponents and Distributive Lattices

54

(3)

Characterization Problems for Posets, Lattices, Graphs, and Families of Sets

Overview

57

(1)

Comparability Graphs and Gallai's Theorem

58

(4)

The 3-Irreducible Posets---Trotter and Moore's Method

62

(4)

Planar Posets and Planar Aographs

66

(3)

Planar Lattices

69

(8)

The 3-Irreducible Posets---Kelly's Method

77

(4)

Stacks and 3-Interval Irreducible Posets

81

(5)

Interval Graphs and Interval Orders

86

(2)

Linear and Interval Families of Sets

88

(2)

Boxes in Rt

90

(2)

Circular Arc Graphs

92

(1)

Characterizing Inequalities

93

(4)

Hypergraph Coloring, Computational Complexity, and Irreducible Posets

Overview

97

(1)

Hypergraph Coloring

98

(4)

Irreducible Posets

102

(6)

Embedding Posets and Kelly's Dimension Products

108

(4)

Counting Irreducible Posets

112

(1)

Planar Posets and Trees

Overview

113

(1)

Planar Posets with a Zero

114

(4)

Planar Posets with Arbitrary Dimension

118

(1)

Repeated Splits and Amalgamations

119

(3)

Chordal Graphs and Cycle-Free Posets

122

(5)

Planar Graphs, Planar Maps, and Convex Polytopes

Overview

127

(1)

Schnyder's Dimension Theoretic Test for Planarity

128

(4)

Convex Polytopes

132

(3)

Normal Families of Paths

135

(7)

Constructing Normal Families of Paths

142

(7)

Convex Polytopes and Irreducible Posets

149

(2)

Planar Multigraphs

151

(6)

Probabilistic Methods in Dimension Theory

Overview

157

(1)

Subposets of the Subset Lattice

158

(7)

Posets of Bounded Degree

165

(1)

The Furedi/Kahn Bounds

166

(2)

Random Bipartite Posets

168

(1)

Upper Bounds

169

(2)

Lower Bounds

171

(10)

The Dimension of a Random Labeled Poset

181

(5)

Winkler's Model for Random Posets

186

(3)

Interval and Geometric Containment Orders

Overview

189

(1)

Interval Orders and Semi-orders

190

(6)

Dimension Theory for Semi-orders and Interval Orders

196

(1)

Shift Graphs

197

(4)

Ramsey Trails

201

(1)

Interval Dimension

202

(2)

Containment Orders

204

(5)

Degrees of Freedom

209

(1)

Circle Orders and Sphere Orders

210

(3)

Greedy Dimension, Back-Tracking, and Depth First Search

Overview

213

(1)

The Jump number Problem

214

(1)

Algorithms for Linear Extensions

215

(2)

Inequalities for Greedy Dimension

217

(7)

Partitions and Constructions

224

(11)

Super-Greedy Extensions, Back-Tracking, and Depth First Search

235

(12)

Products of Chains of Bounded Length

Overview

247

(1)

Embedding Posets in Cubes

247

(2)

Matchings in Posets

249

(2)

Removal Theorems for k-Dimension

251

(5)

Ramsey Theory for Posets

256

(2)

The Maximum Dimension of a Lattice

258

(3)

Large Minimal Realizers

Overview

261

(1)

Minimal Realizers and Critical Digraphs

261

(7)

Applications of the Rank Reformulation Theorem

268

(2)

Generalizing Turan's Theorem

270

(15)

Appendix

285

(2)

Bibliography

287

(14)

Index

301

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Combinatorics and Partially Ordered Sets

Summary

Author Biography

Table of Contents

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Summary

Author Biography

Table of Contents

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