Combinatorics of finite sets. (English) Zbl 0604.05001

Oxford Science Publications. Oxford: Clarendon Press. XV, 250 p. £22.50 (1987).
This book introduces the reader into the techniques and results on the extremal problems of combinatorics. Starting from Sperner’s theorem it not only surveys most of the theorems, but with a very good didactical sense, repeats, generalizes, specializes, and returns (from a different direction) to the original. The reader (any graduate student) may, therefore, not only learn the subject, but also the relative importance of the theorems and methods, such as the Erdős-Ko-Rado theorem or the symmetric chain decomposition.
Chapter One is devoted to Sperner’s theorem with some immediate generalizations. Chapter Two gives Sperner’s original proof, and investigates posets with the LYM-property. The author returns to antichains in Chapters Eight and Thirteen (posets of antichains). Chapter Three treats the symmetric chain decomposition, with applications. Chapters Four, Nine, and Ten give generalizations of different notions to multisets. Chapter Five tells us the Erdős-Ko-Rado theorem with some applications. Chapter Six treats a powerful lemma of Kleitman’s. Chapter Seven is for the Kruskal-Katona theorem. Chapter Eleven treats the Littlewood-Offord problem, and Chapter Twelve gives other results. At the end of the chapters more than 150 problems treat other, sometimes recent, results. The book concludes with a section of hints and solutions to these problems.
Reviewer: P.Komjath


05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05A20 Combinatorial inequalities
05C65 Hypergraphs
05C35 Extremal problems in graph theory