Basics of Ramsey Theory

Dr. Veselin Jungić is a Teaching Professor at the Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada. Dr. Jungić is a 3M National Teaching Fellow and a Fellow of the Canadian Mathematical Society. He is a recipient of several teaching awards. Veselin is one of only a few Canadian mathematicians who has been awarded both the Canadian Mathematical Society Pouliot Award (2020) and the Canadian Mathematical Society Teaching Award (2012). Dr. Jungić’s publications range from education related opinion pieces to articles based on his teaching practices to Ramsey theory research and outreach papers. One of Dr. Jungić’s accomplishments is the creation of the Math Catcher Outreach Program. Since the early 2010s, the Program has visited hundreds of classrooms, from kindergarten to grade 12, and created learning resources in multiple Indigenous languages. As an invited speaker, Veselin delivered several dozens of the Math Catcher–related workshops and lectures to teachers, academics, and public at the local, national, and international levels.

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.