In the preface to the first edition of his book, Ian Anderson wrote in 1985,” The past quarter century has seen the remarkable rise of combinatorics as a distinctive and important area of mathematics.” It is no less true after more than a quarter century.

His book, as much a set theory book as one on combinatorics, is something of a throwback. While virtually every combinatorics text strives to include copious applications, Anderson’s main focus is the theory. For example, an overarching theme of the text is Sperner’s theorem (in various contexts) and its generalizations. Sperner’s Theorem, as you may recall, is concerned with the cardinality of a collection of subsets of a certain size with the property that no members of the collection are subsets of each other.

Anderson presents several proofs of Sperner’s Theorem, as well as applications of it. For example, he demonstrates its importance in the development of the notion of partially ordered sets. This then leads to a rather elegant treatment of posets and Stirling numbers. In addition, Anderson develops a probabilistic version of the theorem.

While applications are not the main focus, they are nonetheless covered. In the thirteen chapters of the text, Anderson treats such important combinatorial theorems as Erdős-Ko-Rado and Kruskal-Katona, while also including multisets, lattices, and even some coverage of simplicial complexes, graph theory, and networks as posets. Obviously, a background in abstract algebra as well as discrete mathematics is necessary for a true appreciation of this text.

There are numerous exercises, with hints and solutions for many of them included. The development of the main ideas utilizes first principles. That, coupled with the presentation of alternate proofs of Sperner’s and other theorems, make it ideal for discussing the aesthetic of proof. This text is excellent for advanced undergraduates and beginning graduate students.

John T. Saccoman is Professor of Mathematics at Seton Hall University in South Orange, NJ.