Ramsey理论是对数学对象的结构的研究，这本创新的书提供了Ramsey理论对整数的第一个有凝聚力的研究。它可能包含了这个蓬勃发展的学科中已解决和未解决问题的最实质性的说明。本书适合对组合学、数论和Ramsey理论感兴趣的研究生和数学研究人员阅读参考。 Ramsey theory is the study of the structure of mathematical objects that is preserved under partitions. In its full generality, Ramsey theory is quite powerful, but can quickly become complicated. By limiting the focus of this book to Ramsey theory applied to the set of integers, the authors have produced a gentle, but meaningful, introduction to an important and enticing branch of modern mathematics. Ramsey Theory on the Integers offers students a glimpse into the world of mathematical research and the opportunity for them to begin pondering unsolved problems. For this new edition, several sections have been added and others have been significantly updated. Among the newly introduced topics are: 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 Erd?s-Ginzberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colorings.

前辅文
Preface to the Second Edition
Acknowledgements
Preface to the First Edition
Chapter 1. Preliminaries
  §1.1. The Pigeonhole Principle
  §1.2. Ramsey's Theorem
  §1.3. Some Notation
  §1.4. Three Classical Theorems
  §1.5. A Little More Notation
  §1.6. Exercises
  §1.7. Research Problems
  §1.8. References
Chapter 2. Van der Waerden's Theorem
  §2.1. The Compactness Principle
  §2.2. Alternate Forms of van der Waerden's Theorem
  §2.3. Computing van der Waerden Numbers
  §2.4. Bounds on van der Waerden Numbers
  §2.5. The Erdós and Turán Function
  §2.6. On the Number of Monochromatic Arithmetic Progressions
  §2.7. Proof of van der Waerden's Theorem
  §2.8. Exercises
  §2.9. Research Problems
  §2.10. References
Chapter 3. Supersets of AP
  §3.1. Quasi-Progressions
  §3.2. Generalized Quasi-Progressions
  §3.3. Descending Waves
  §3.4. Semi-Progressions
  §3.5. Iterated Polynomials
  §3.6. Arithmetic Progressions as Recurrence Solutions
  §3.7. Exercises
  §3.8. Research Problems
  §3.9. References
Chapter 4. Subsets of AP
  §4.1. Finite Gap Sets
  §4.2. Infinite Gap Sets
  §4.3. Exercises
  §4.4. Research Problems
  §4.5. References
Chapter 5. Other Generalizations of w(k;r)
  §5.1. Sequences of Typе x, ax+d, bx+2d
  §5.2. Homothetic Copies of Sequences
  §5.3. Sequences of Typе x, x+d, x+2d+b
  §5.4. Polynomial Progressions
  §5.5. Exercises
  §5.6. Research Problems
  §5.7. References
Chapter 6. Arithmetic Progressions (modm)
  §6.1. The Family of Arithmetic Progressions (modm)
  §6.2. A Seemingly Smaller Family is More Regular
  §6.3. The Degree of Regularity
  §6.4. Exercises
  §6.5. Research Problems
  §6.6. References
Chapter 7. Other Variations on van der Waerden's Theorem
  §7.1. The Function Γm(k:)
  §7.2. Monochromatic Sets a(S+6)
  §7.3. Having Most Elements Monochromatic
  §7.4. Permutations Avoiding Arithmetic Progressions
  §7.5. Exercises
  §7.6. Research Problems
  §7.7. References
Chapter 8. Schur's Theorem
  §8.1. The Basic Theorem
  §8.2. A Generalization of Schur's Theorem
  §8.3. Refinements of Schur's Theorem
  §8.4. Schur Inequality
  §8.5. Exercises
  §8.6. Research Problems
  §8.7. References
Chapter 9. Rado's Theorem
  §9.1. Rado's Single Equation Theorem
  §9.2. Some Rado Numbers
  §9.3. Generalizations of the Single Equation Theorem
  §9.4. Solutions to Linear Recurrences
  §9.5. Mixing Addition and Multiplication
  §9.6. Exercises
  §9.7. Research Problems
  §9.8. References
Chapter 10. Other Topics
  §10.1. Monochromatic Sums
  §10.2. Doublefree Sets
  §10.3. Diffsequences
  §10.4. Brown's Lemma
  §10.5. Monochromatic Sets Free of Prescribed Differences
  §10.6. Patterns in Colorings
  §10.7. Rainbow Ramsey Theory on the Integers
  §10.8. Zero-Sums and m-Sets
  §10.9. Exercises
  §10.10. Research Problems
  §10.11. References
Notation
Bibliography
Index