Basics of Ramsey Theory

1. Introduction: Pioneers and Trailblazers. 1.1. Complete Disorder is Impossible. 1.2 Paul Erdos. 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. Erdos-Szekeres' Upper and Lower Bounds. 6.4. Progress on the Conjecture OF Erdos and Szekeres. 6.5. Exercises. 7. Solutions.