Sperner theory

Cover ......Page 1
Title ......Page 5
Contents ......Page 7
Preface ......Page 9
1.1 Spemer\'s theorem ......Page 13
1.2 Notation and tenninology ......Page 16
1.3 The main examples ......Page 21
2.1 Counting in two different ways ......Page 28
2.2 Partitions into symmetric chains ......Page 41
2.3 Exchange operations and compression ......Page 45
2.4 Generating families ......Page 62
2.5 Linear independence ......Page 73
2.6 Probabilistic methods ......Page 83
3 Profile-polytopes for set families ......Page 96
3.1 Full hereditary families and the antiblocking type ......Page 98
3.2 Reduction to the circle ......Page 102
3.3 Classes of families arising from Boolean expressions ......Page 105
4 The flow-theoretic approach in Spemer theory ......Page 128
4.1 The Max-Flow Min-Cut Theorem and the Min-Cost Flow Algorithm ......Page 129
4.2 The k-cutset problem ......Page 137
4.3 The k-family problem and related problems ......Page 143
4.4 The variance problem ......Page 152
4.5 Nonnal posets and flow morphisms ......Page 160
4.6 Product theorems ......Page 178
5.1 Definitions, main properties, and examples ......Page 191
5.2 More part Spemer theorems and the Littlewood-Offord problem ......Page 199
5.3 Coverings by intervals and sc-orders ......Page 206
5.4 Semisymmetric chain orders and matchings ......Page 210
6 Algebraic methods in Spemer theory ......Page 220
6.1 The full rank property and Jordan functions ......Page 221
6.2 Peck posets and the commutation relation ......Page 241
6.3 Results for modular, geometric, and distributive lattices ......Page 260
6.4 The independence number of graphs and the Erdos-Ko-Rado Theorem ......Page 288
6.5 Further algebraic methods to prove intersection theorems ......Page 307
7.1 Central and local limit theorems ......Page 316
7.2 Optimal representations and limit Spemer theorems ......Page 329
7.3 An asymptotic Erdos-Ko-Rado Theorem ......Page 340
8 Macaulay po sets ......Page 344
8.1 Macaulay posets and shadow minimization ......Page 345
8.2 Existence theorems for Macaulay posets ......Page 363
8.3 Optimization problems for Macaulay po sets ......Page 368
8.4 Some further numerical and existence results for chain products ......Page 379
8.5 Spemer families satisfying additional conditions in chain products ......Page 390
Notation ......Page 402
Bibliography ......Page 407
Index ......Page 425