Basics 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 Hales–Jewett 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) accessible to undergraduate students
- Serves as both a textbook or as a supplementary text in an elective course in combinatorics and aimed at a diverse group of students interested in mathematics
This book 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.
1. Introduction: Pioneers and Trailblazers. 1.1. Complete Disorder is
Impossible. 1.2 Paul Erds. 1.3. Frank Plumpton Ramsey. 1.4 Ramsey Theory.
2.
Ramseys Theorem. 2.1. The Pigeonhole Principle. 2.2. Acquaintances and
Strangers. 2.3. Ramseys Theorem for Graphs. 2.4. Ramseys Theorem: Infinite
Case. 2.5. Ramseys Theorem: General Case. 2.6. Exercises.
3. van der
Waerdens Theorem. 3.1. Bartel van der Waerden. 3.2. van der Waerdens
Theorem: 3Term Arithmetic Progressions. 3.3. Proof of van der Waerdens
Theorem. 3.4. van der Waerdens Theorem: How Far and Where? 3.5. van der
Waerdens Theorem: Some Related Questions. 3.6. Exercises.
4. Schurs Theorem
and Rados Theorem. 4.1 Issai Schur. 4.2. Schurs Theorem. 4.3. Richard Rado.
4.4 Rados Theorem. 4.5. Exercises.
5. The HalesJewett Theorem. 5.1.
Combinatorial Lines. 5.2. Generalized TicTacToe Game. 5.3. The HalesJewett
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. ErdsSzekeres Upper and Lower Bounds. 6.4. Progress on
the Conjecture OF Erds and Szekeres. 6.5. Exercises.
7. Solutions.
Dr. Veselin Jungi is a Teaching Professor at the Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada.
Dr. Jungi 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. Jungis publications range from education related opinion pieces to articles based on his teaching practices to Ramsey theory research and outreach papers.
One of Dr. Jungis 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 Catcherrelated workshops and lectures to teachers, academics, and public at the local, national, and international levels.
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