Fundamentals of Ramsey Theory

1. Introduction. 1.1. What is Ramsey Theory? 1.2. Notations and Conventions. 1.3. Prerequisites. 1.4. Compactness Principle. 1.5. Set Theoretic Considerations. 1.6. Exercises. 2. Integer Ramsey Theory. 2.1. Van der Waerden's Theorem. 2.2. Equations. 2.3. Hales-Jewett Theorem. 2.4. Finite Sums. 2.5. Density Results. 2.6. Exercises. 3. Graph Ramsey Theory. 3.1. Complete Graphs. 3.2. Other Graphs. 3.3. Hypergraphs. 3.4. Infinite Graphs. 3.5. Comparing Ramsey and van der Waerden Results. 3.6. Exercises. 4. Euclidean Ramsey Theory. 4.1. Polygons. 4.2. Chromatic Number of the Plane. 4.3. Four Color Map Theorem. 4.4. Exercises. 5. Other Approaches to Ramsey Theory. 5.1. Topological Approaches. 5.2. Ergodic Theory. 5.3. Stone-Cech Compactification. 5.4. Additive Combinatorics Methods. 5.5. Exercises. 6. The Probabilistic Method. 6.1. Lower Bounds on Ramsey, van der Waerden, and Hales-Jewett Numbers. 6.2. Turan's Theorem. 6.3. Almost-surely van der Waerden and Ramsey Numbers. 6.4. Lovasz Local Lemma. 6.5. Exercises. 7. Applications. 7.1. Fermat's Last Theorem. 7.2. Encoding Information. 7.3. Data Mining. 7.4. Exercises. Bibliography. Index.

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Ramsey Theory;Arithmetic Progression;combinatorics;Indicator Random Variable;van der Waerden's theorem;Ramsey Numbers;partition regularity;Compactness Principle;feermat's last theorem;Independent Set;hales-jewett theorem;Planar Graph;Largest Independent Set;Ergodic systems;Dependency Graph;Topological dynamics;Threshold Graph;Ramsey-type numbers;Chromatic Number;Euclidean Ramsey theory;Dual Graph;Basic Probabilistic Method;Monochromatic Copy;Side Length;Equilateral Triangle;Pigeonhole Principle;Color Theorem;Discrete Fourier Transform;Van Der Waerden;Red Neighbors;Regular Tetrahedron;Inclusion Exclusion Formula;Noisy Channel;Stereographic Projection