In 1986, when this reviewer was a freshman in college, he got and read a copy of Promel’s identically titled habilitation dissertation. This was the first work of that level that the reviewer had ever seen. Therefore he was very curious to learn how the topic had evolved during the 28 years that have passed since.
It turns out that the book has changed relatively little. Hans Jürgen Prömel got more and more involved in university administration, which culminated in his becoming the president of the University of Darmstadt, Germany. As his 60th birthday was approaching, a few of his friends, led by Angelica Steger, decided to turn his Habilitationschrift into a book. This is the book that Springer published and is under review here.
Parts I–III of the book (Basics, Parameter Sets, and Sets) are as they were 28 years ago. They cover, among other things, Ramsey numbers and their upper and lower bounds, Van der Waerden’s theorem, and the Hales-Jewett theorem. Parts IV and V of the old book have been merged into one, and are complemented by material on random Ramsey theory. Finally, Part V of the present book is new, and it focuses on the combinatorial proof of the density Hales-Jewett theorem.
If you are not an expert on current affairs in Discrete Ramsey theory, the book will give you a thorough introduction to the classic results.
Miklós Bóna is Professor of Mathematics at the University of Florida.