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1st Edition

Extremal Finite Set Theory




ISBN 9781138197848
Published October 9, 2018 by Chapman and Hall/CRC
336 Pages

 
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Book Description

Extremal Finite Set Theory surveys old and new results in the area of extremal set system theory. It presents an overview of the main techniques and tools (shifting, the cycle method, profile polytopes, incidence matrices, flag algebras, etc.) used in the different subtopics. The book focuses on the cardinality of a family of sets satisfying certain combinatorial properties. It covers recent progress in the subject of set systems and extremal combinatorics.

Intended for graduate students, instructors teaching extremal combinatorics and researchers, this book serves as a sound introduction to the theory of extremal set systems. In each of the topics covered, the text introduces the basic tools used in the literature. Every chapter provides detailed proofs of the most important results and some of the most recent ones, while the proofs of some other theorems are posted as exercises with hints.

Features:

  • Presents the most basic theorems on extremal set systems
  • Includes many proof techniques
  • Contains recent developments
  • The book’s contents are well suited to form the syllabus for an introductory course

About the Authors:

Dániel Gerbner is a researcher at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences in Budapest, Hungary. He holds a Ph.D. from Eötvös Loránd University, Hungary and has contributed to numerous publications. His research interests are in extremal combinatorics and search theory.

Balázs Patkós is also a researcher at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences. He holds a Ph.D. from Central European University, Budapest and has authored several research papers. His research interests are in extremal and probabilistic combinatorics.

Table of Contents

Basics

Sperner’s theorem, LYM-inequality, Bollobás inequality. The Erdős-Ko-Rado theorem - several proofs. Intersecting Sperner families. Isoperimetric inequalities: the Kruskal-Katona theorem and Harper’s theorem. Sunflowers.

Intersection theorems

Stability of the Erdős-Ko-Rado theorem. t-intersecting families. Above the Erdős-Ko-Rado threshold. L-intersecting families. r-wise intersecting families. k-uniform intersecting families with covering number k. The number of intersecting families. Cross-intersecting families.

Sperner-type theorems

More-part Sperner families. Supersaturation. The number of antichains in 2^{[n]} (Dedekind’s problem). Union-free families and related problems. Union-closed families.

Random versions of Sperner’s theorem and the Erdős-Ko-Rado theorem

The largest antichain in Qn (p). Largest intersecting families in Qn, k (p). Removing edges from K n (n, K). G-intersecting families. A random process generating intersecting families.

Turán-type problems

Complete forbidden hypergraphs and local sparsity. Graph-based forbidden hypergraphs. Hypergraph-based forbidden hypergraphs. Other forbidden hypergraphs. Some methods. Non-uniform Turán problems

Saturation problems

Saturated hypergraphs and weak saturation. Saturating k-Sperner families and related problems.

Forbidden subposet problems

Chain partitioning and other methods. General bounds on La(n, P) involving the height of P. Supersaturation. Induced forbidden subposet problems. Other variants of the problem. Counting other subposets.

Traces of sets

Characterizing the case of equality in the Sauer Lemma. The arrow relation. Forbidden subconfigurations. Uniform versions.

Combinatorial search theory

Basics. Searching with small query sets. Parity search. Searching with lies. Between adaptive and non-adaptive algorithms

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Author(s)

Biography

Dániel Gerbner is a researcher at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences in Budapest, Hungary. He holds a Ph.D. from Eötvös Loránd University, Hungary and has contributed to numerous publications. His research interests are in extremal combinatorics and search theory.

Balázs Patkós is also a researcher at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences. He holds a Ph.D. from Central European University, Budapest and has authored several research papers. His research interests are in extremal and probabilistic combinatorics.