Extremal Finite Set Theory

Extremal Finite Set Theory is a book that surveys old and new results in extremal set system theory, focusing on the cardinality of a family of sets satisfying certain combinatorial properties. It is intended for graduate students, instructors teaching extremal combinatorics, and researchers, and provides a sound introduction to the theory.

Format: Paperback / softback
Length: 352 pages
Publication date: 21 January 2023
Publisher: Taylor & Francis Ltd

Extremal Finite Set Theory is a comprehensive survey of both established and recent developments in the field of extremal set system theory. It provides an overview of the primary techniques and tools employed across various subtopics, including shifting, the cycle method, profile polytopes, incidence matrices, flag algebras, and more. The book's central focus lies in the study of the cardinality of a family of sets that satisfy specific combinatorial properties. It encompasses recent advancements in set systems and extremal combinatorics, making it an invaluable resource for graduate students, instructors, and researchers alike.

The text is structured into chapters, each dedicated to a specific topic within extremal set systems. Within each chapter, the fundamental tools used in the literature are introduced. Detailed proofs of the most important results are provided, along with some recent developments. Additionally, certain theorems are presented as exercises with hints, encouraging readers to engage with the material actively.

One of the notable features of this book is its emphasis on presenting the most basic theorems on extremal set systems. These theorems serve as the foundation upon which subsequent developments are built. The text employs various proof techniques to demonstrate the validity of these results, making it accessible to readers with a background in mathematics.

Furthermore, the book incorporates recent developments in the field, reflecting the ongoing progress in extremal set system theory. This includes discussions on the connections between extremal set systems and other areas such as graph theory, probability theory, and combinatorics.

The contents of this book are well-suited to form the syllabus. syllabus for an introductory course in extremal set systems. The authors, Dániel Gerbner and Balázs Patkós, are both researchers at the Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, in Budapest, Hungary. Gerbner holds a Ph.D. from Eötvös Loránd University, Hungary, and has contributed extensively to the field of extremal combinatorics. Patkós holds a Ph.D. from Central European University, Budapest, and has authored several research papers in extremal and probabilistic combinatorics.

In conclusion, Extremal Finite Set Theory is an essential resource for anyone interested in exploring the theory of extremal set systems. It provides a comprehensive and up-to-date overview of the field, covering both established results and recent advancements. With its clear explanations, detailed proofs, and exercises, the book serves as a valuable introduction to graduate students, instructors, and researchers alike.

Weight: 650g
Dimension: 234 x 156 (mm)
ISBN-13: 9781032476001