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Ramsey Theory on the Integers

Bruce M. Landman and Aaron Robertson
Publisher: 
American Mathematical Society
Publication Date: 
2014
Number of Pages: 
384
Format: 
Paperback
Edition: 
2
Series: 
Student Mathematical Library 73
Price: 
61.00
ISBN: 
9780821898673
Category: 
Monograph
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on
11/20/2014
]

See our review of the first edition, which was published in 2003.

The authors note that one of the goals of the first edition was to stimulate research on Ramsey theory on the integers. It appears that they have been successful in that regard, because many of the research problems mentioned in that edition have now been solved! As a result, the new edition includes many changes, including sections on recent advances. New topics include

… rainbow Ramsey theory, an “inequality” version of Schur’s theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdös-Ginsberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colorings. We also offer many new results and proofs among the topics that are not new to this edition, most of which were not known when the first edition was published.

In other words, an update was much needed, and now it has been done. This is an excellent book on a fascinating topic.

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Fernando Q. Gouvêa is the Editor of MAA Reviews.

  • Preliminaries
  • Van der Waerden's theorem
  • Supersets of AP
  • Subsets of AP
  • Other generalizations of w(k;r)
  • Arithmetic progressions (modm)
  • Other variations on van der Waerden's theorem
  • Schur's theorem
  • Rado's theorem
  • Other topics
  • Notation
  • Bibliography
  • Index