Extremal Finite Set Theory

Extremal Finite Set Theory surveys old and new results in this subject. It presents an overview of the main techniques and tools (shifting, cycle method, profile polytopes, incidence matrices, combinatorial nullstellensatz, etc.) used in the different subtopics.

Autorentext

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.

Klappentext

Extremal Finite Set Theory surveys old and new results in this subject. It presents an overview of the main techniques and tools (shifting, cycle method, profile polytopes, incidence matrices, combinatorial nullstellensatz, etc.) used in the different subtopics.

Zusammenfassung
Extremal Finite Set Theory surveys old and new results in this subject. It presents an overview of the main techniques and tools (shifting, cycle method, profile polytopes, incidence matrices, combinatorial nullstellensatz, etc.) used in the different subtopics.

Inhalt

Basics

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

Intersection theorems

Stability of the Erd s-Ko-Rado theorem. t-intersecting families. Above the Erd s-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 Erd s-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