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.
Basics
Sperners theorem, LYM-inequality, Bollobás inequality. The Erdos-Ko-Rado theorem - several proofs. Intersecting Sperner families. Isoperimetric inequalities: the Kruskal-Katona theorem and Harpers theorem. Sunflowers.
Intersection theorems
Stability of the Erdos-Ko-Rado theorem. t-intersecting families. Above the Erdos-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]} (Dedekinds problem). Union-free families and related problems. Union-closed families.
Random versions of Sperners theorem and the Erdos-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
