Extremal Finite Set Theory

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.

• BasicsSperner’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 theoremsStability 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 theoremsMore-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 theoremThe 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 problemsComplete forbidden hypergraphs and local sparsity. Graph-based forbidden hypergraphs. Hypergraph-based forbidden hypergraphs. Other forbidden hypergraphs. Some methods. Non-uniform Turán problemsSaturation problemsSaturated hypergraphs and weak saturation. Saturating k-Sperner families and related problems.Forbidden subposet problemsChain 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 setsCharacterizing the case of equality in the Sauer Lemma. The arrow relation. Forbidden subconfigurations. Uniform versions.Combinatorial search theoryBasics. Searching with small query sets. Parity search. Searching with lies. Between adaptive and non-adaptive algorithms

"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 oered 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