Description
Book SynopsisBasics of Ramsey Theory serves as a gentle introduction to Ramsey theory for students interested in becoming familiar with a dynamic segment of contemporary mathematics that combines ideas from number theory and combinatorics. The core of the of the book consists of discussions and proofs of the results now universally known as Ramsey's theorem, van der Waerden's theorem, Schur's theorem, Rado's theorem, the HalesJewett theorem, and the Happy End Problem of Erdos and Szekeres. The aim is to present these in a manner that will be challenging but enjoyable, and broadly accessible to anyone with a genuine interest in mathematics.
Features
- Suitable for any undergraduate student who has successfully completed the standard calculus sequence of courses and a standard first (or second) year linear algebra course
- Filled with visual proofs of fundamental theorems
- Contains numerous exercises (with their solutions) acce
Table of Contents
1. Introduction: Pioneers and Trailblazers. 1.1. Complete Disorder is Impossible. 1.2 Paul Erdős. 1.3. Frank Plumpton Ramsey. 1.4 Ramsey Theory. 2. Ramsey’s Theorem. 2.1. The Pigeonhole Principle. 2.2. Acquaintances and Strangers. 2.3. Ramsey’s Theorem for Graphs. 2.4. Ramsey’s Theorem: Infinite Case. 2.5. Ramsey’s Theorem: General Case. 2.6. Exercises. 3. van der Waerden’s Theorem. 3.1. Bartel van der Waerden. 3.2. van der Waerden’s Theorem: 3–Term Arithmetic Progressions. 3.3. Proof of van der Waerden’s Theorem. 3.4. van der Waerden’s Theorem: How Far and Where? 3.5. van der Waerden’s Theorem: Some Related Questions. 3.6. Exercises. 4. Schur’s Theorem and Rado’s Theorem. 4.1 Issai Schur. 4.2. Schur’s Theorem. 4.3. Richard Rado. 4.4 Rado’s Theorem. 4.5. Exercises. 5. The Hales–Jewett Theorem. 5.1. Combinatorial Lines. 5.2. Generalized Tic–Tac–Toe Game. 5.3. The Hales–Jewett Theorem. 5.4. Exercises. 6. Happy End Problem. 6.1. The Happy End Problem: Triangles, Quadrilaterals, and Pentagons. 6.2. The Happy End Problem – General Case. 6.3. Erdős–Szekeres’ Upper and Lower Bounds. 6.4. Progress on the Conjecture OF Erdős and Szekeres. 6.5. Exercises. 7. Solutions.
