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Positional games. (English) Zbl 1314.91003

Oberwolfach Seminars 44. Basel: Birkhäuser/Springer (ISBN 978-3-0348-0824-8/pbk; 978-3-0348-0825-5/ebook). x, 146 p. (2014).
The blurb on the back cover provides a good description of the contents of the book, here are some additional comments. The book is on positional games, an active area of combinatorics at the moment. It is centred around maker-breaker games, their special cases (e.g., the connectivity game, the Hamiltonicity game, and graphs with bounded hyperedge intersection), and their variants (e.g., assuming one player can do several moves in their turn, assuming one player wants to avoid a structure while the opponent wants to enforce it, assuming the board has some random elements deleted from it, random players, and so on).
There are a few things I like about this book:
1. It covers many (if not all) of the recent advances in the area that have appeared during 2010–2014.
2. It is very well written: gives a lot of intuition about the proofs; it breaks the proofs into small, digestible pieces, omits more technical proofs by referring to the original papers, motivates the defined concepts, etc.
3. Mathematical maturity and basic combinatorics and probability is all one needs to know for reading the book: all necessary concepts are defined. Hence it is accessible to most mathematics students/researchers.
4. The chapters are short and “atomic”, the whole book is also not too long, and each chapter has exercises, making it an ideal classroom textbook.
5. Several nice connections are made with other areas of mathematics, in particular random graphs and randomized algorithms.
6. The book can also be read, as a comprehensive survey, by anyone who wants to work on positional games, and wants to know the main problems, techniques and results in the area.
Publisher’s description: This text is based on a lecture course given by the authors in the framework of Oberwolfach Seminars at the Mathematisches Forschungsinstitut Oberwolfach in May, 2013. It is intended to serve as a thorough introduction to the rapidly developing field of positional games. This area constitutes an important branch of combinatorics, whose aim it is to systematically develop an extensive mathematical basis for a variety of two player perfect information games. These ranges from such popular games as Tic-Tac-Toe and Hex to purely abstract games played on graphs and hypergraphs. The subject of positional games is strongly related to several other branches of combinatorics such as Ramsey theory, extremal graph and set theory, and the probabilistic method. These notes cover a variety of topics in positional games, including both classical results and recent important developments. They are presented in an accessible way and are accompanied by exercises of varying difficulty, helping the reader to better understand the theory. The text will benefit both researchers and graduate students in combinatorics and adjacent fields.

MSC:

91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91A24 Positional games (pursuit and evasion, etc.)
91A80 Applications of game theory
05C51 Graph designs and isomorphic decomposition
05C80 Random graphs (graph-theoretic aspects)
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
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