Introduction to Combinatorial Designs

This is an introductory graduate textbook that is very easy to read. There are almost no prerequisites: a little bit of linear algebra, the notion of congruence, and some counting principles up to the notion of binomial coefficients, so nothing that a fourth-year undergraduate majoring in mathematics would not know. Therefore, the book is perfect for a reading course taken by an ambitious undergraduate. It is also a useful reference material for the non-specialist researcher needing basic information on designs.

Besides the classic theory, there are several sections devoted to applications. The first one is on one-factorizations, which includes the scheduling round-robin tournaments. We learn, for instance, that in any round-robin tournament of 2n teams, in which each team plays every other team twice, once home and once away from home, there will be at most two teams that do not have to play home twice in a row, or away from home twice in a row. We also learn that it is possible to schedule such a tournament, though, in a way that each of the remaining 2n–2 teams will only have to play two such consecutive games once. There are other applications, to statistics and cryptography, but in less detail. In the experience of this reviewer, students interested in designs are always very interested in error-correcting codes, and this book devotes less than two pages to them.

There is a sufficient number of exercises, but very few of them come with complete solutions. Students of this reviewer always complain about books lacking them.

Other than that, my only critical remark is that the book does not discuss nearly enough open problems. When an advanced mathematics textbook is all about facts and not about questions, it can leave the false impression that its subject is a finished discipline. That can discourage innovative students from going into the field. That said, instructors looking for a textbook for a basic graduate course on designs should certainly consider this very reader-friendly volume.

Miklós Bóna is Associate Professor of Mathematics at the University of Florida.