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Research problems in discrete geometry. (English) Zbl 1086.52001
New York, NY: Springer (ISBN 0-387-23815-8/hbk). xii, 499 p. (2005).
Paul Erdős asked me once whether I read “The Book.” “Of course not,” answered I, thinking that Paul referred, as he often did, to God’s created “transfinite book that contained all problems and their best solutions.” “Call [Willy] Moser, he will send you a copy,” Erdős said. This is how I received my first of many versions of “Research Problems in Discrete Geometry.”
The history of The Book deserves its own book. The famous problem poser and solver Leo Moser started it 1963, with his own 50 open problems. The flame was kept alive by his younger brother and coauthor Willy Moser, who was joined by János Pach in 1986, and by Peter Brass in 2000. Looking at many versions of The Book in front of me, I see how wonderfully the book has grown. It now includes over 500 open problems of discrete geometry, many more than any book on the subject that I know. Yet, The Book retains the attraction of its earlier versions: the problems are very contagious, easy to understand, yet hard to solve, especially to solve completely.
Contents include Density Problems for Packings and Coverings; Structural Packing and Covering Problems; Packing and Covering with Homothetic Copies; Tiling Problems; Distance Problems; Problems on Repeated Subconfigurations; Incidence and Arrangement Problems; Problems on Points in General Position; Graph Drawing and Geometric Graphs; Lattice Point Problems; and Geometric Inequalities.
At the age of computing and information, it is only natural that discrete mathematics is flourishing. It is therefore natural that the authors could not include all aspect of discrete geometry in one book. They chose areas close to them, and left to others to write about such exciting areas of discrete geometry as convex polytopes, Helly-type problems, stochastic geometry, etc.
The book is written in the style of Paul Erdős’s problem articles. It is fitting therefore that The Book includes Paul’s introduction written for an earlier version. Each section is a self-contained essay that provides the history of a problem, known partial results, motivation and bibliography – everything needed for the reader, especially a young mathematician seeking a problem to conquer, to start his or her research. Professionals will find The Book to be a valuable encyclopedic source for their research. Indeed, this is an exceptional book, a must for all active mathematicians.
The review would be incomplete without a sample problem. I choose, quite subjectively, my favorite unsolved problem in all of mathematics (Problem 1, section 5.9): What is the minimum number of colors for coloring the points of the plane so that no two points at unit distance receive the same color?
The last words of Paul Erdős’s introduction are: “I wish the reader good luck with the solutions!” I can only add, Amen.

MSC:
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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