Extremal Finite Set Theory: (Discrete Mathematics and Its Applications)

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

About the Author :
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.

Review :
"Materials collected in this book were meant to be presented at the Extremal Sets Systems Seminar of the Alfréd Rényi Institute of Mathematics. The main focus is placed on maximizing the cardinality of a family of sets satisfying some prescribed properties. The authors pay great attention to present a large range of proof techniques used in dealing with finite sets systems and give an extensive survey of recent results. Although ordinary graphs are 2-uniform families, extremal graph theory is outside the scope of this book. Ramsey-type theorems and coding theory are also avoided. The intended audience includes (1) students eager to learn the central topics on a graduate level; (2) university professors offering courses on extremal combinatorics; (3) researchers refreshing their knowledge and learning recent developments in the field. The book concludes with a comprehensive bibliography of 556 items. This makes it a convenient reference book for people interested in the latest developments in extremal finite set theory." Ko-Wei Lih (Taipei) zbMath September 2019 "This book gives a well-organized and timely summary of research directions in classical extremal combinatorics. As the title clari es, the focus excludes in nitary combinatorics. And, by the authors' own admission, results particular to extremal graph theory or coding theory are omitted. Instead, a considerable level of depth is o ered on the status of the primary topics of extremal set theory: intersection theorems, Sperner families, Tur□an problems, and saturation variants of these. The text is appropriate for graduate students or keen undergrads, and serves equally well as a comprehensive reference for researchers. The book has over 500 insightful exercises which build on the presentation. These are especially valuable for young researchers in the subject, who can gain familiarity with the methods and connect various topics." ~Peter James Dukes--Mathematical Reviews Oct. 2019