El. knyga: Mathematics of Ramsey Theory

Ramsey Theory Old and New.-
1. Ramsey Numbers.-
2. Transfinite Ramsey
Theory.-
3. Chromatic Number.-
4. Classical Theorems.-
5. Other Classical
Theorems.-
6. Structural Generalizations.-
7. Infinite Ramsey Theorem.-
8.
Unprovability Results.-
9. Non-Standard Applications.- I. Classics.- Problems
and Results on Graphs and Hypergraphs: Similarities and Differences.-
1.
Extremal Problems of Turan Type.-
2. Density Problems.-
3. Ramsey's Theorem.-
4. Ramsey-Turan Type Problems.-
5. Chromatic Numbers.- Note on Canonical
Partitions.- II. Numbers.- On Size Ramsey Number of Paths, Trees and
Circuits. II.-
1. Introduction.-
2. Proof of Theorem 1 - Part One.-
3. Proof
of Theorem 1 - Part Two.-
4. Proof of Theorem 2.-
5. Proof of Theorem 3.- On
the Computational Complexity of Ramsey-Type Problems.-
1. Introduction.-
2.
NP-Complete Ramsey Problems.-
3. Polynomial-Bounded Ramsey Problems.-
4.
Discussion.- Constructive Ramsey Bounds and Intersection Theorems for Sets.-
1. Introduction.-
2. Families of Sets with Prescribed Intersections.-
3. The
Proof of Theorem 2.3.-
4. The Proof of Theorem 1.1.- Ordinal Types in Ramsey
Theory and Well-Partial-Ordering Theory.-
1. Introduction.-
2. Sheaves.-
3.
Ramsey Systems.-
4. Well-Partial-Ordering.-
5. Erdos-Szekeres Theorem.-
6.
Ramsey Systems.-
7. Canonical Ramsey Theorem.- III. Structural Theory.-
Partite Construction and Ramsey Space Systems.-
1. Introduction.-
2.
Statement of Results.-
3. Partite Lemma.-
4. Partite Construction.-
5.
Applications.- Graham-Rothschild Parameter Sets.-
1. Introduction.-
2.
Parameter Sets and Parameter Words (Definition and Basic Examples).-
3.
Hales-Jewett's Theorem.-
4. Graham-Rothschild's Theorem.-
5. Infinite
Versions.-
6. Other Structures.- Shelah's Proof of the Hales-Jewett Theorem.-
IV. Noncombinatorial Methods.- Partitioning Topological Spaces.-
1.
Introduction.-
2. Partitioning Singletons.-
3. Partitioning Pairs.-
4. Open
Problems.- Topological Ramsey Theory.-
1. Introduction.-
2. Ramsey Spaces and
Ellentuck's Theorem.-
3. Finitary Consequences of Ellentuck Type Theorems.-
4. The Axiom of Choice and the Construction of Non-Ramsey Sets.-
5. Finite
Dimensional Analogues of Ellentuck Type Theorems.-
6. Canonical Partitions.-
Ergodic Theory and Configurations in Sets of Positive Density.-
1.
Introduction.-
2. Correspondence Between Subsets of ?2 and ?2-Actions.-
3.
Ergodic Averages for Subsets of ?2.-
4. First Application to Subsets of
Positive Density in ?2.-
5. Proof of Theorem A.-
6. A Recurrence Property of
?2-Actions.-
7. Proof of Theorem B.- V. Variations and Applications.- Topics
in Euclidean Ramsey Theory.-
1. Introduction.-
2. Preliminaries.-
3. Ramsey
Sets.-
4. Sphere-Ramsey Sets.-
5. Concluding Remarks.- On Pisier Type
Problems and Results (Combinatorial Applications to Number Theory).-
1.
Introduction.-
2. Multiplicative Bases and Szemeredi-Ruzsa Theorem.-
3.
Graphical Sequences and Examples of Their Use.-
4. Pisier Type Theorems.-
5.
Pisier Problem - Positive Results.-
6. Special Ramsey Graphs - the Partite
Construction.- Combinatorial Statements Independent of Arithmetic.-
1.
Introduction.-
2. Notation.-
3. Arithmetic.-
4. Conclusions.- Boolean
Complexity and Ramsey Theorems.-
1. General Remarks.-
2. An Example of a
Lower Bound to Formula Size Complexity.- Uncrowded Graphs.-
1. Graphs.-
2.
Hypergraphs.-
3. Heilbronn's Conjecture.- Author Index.