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    Ramsey Theory Yesterday, Today, and Tomorrow

    ISBN-10: 0817680918
    ISBN-13: 9780817680916
    Author(s): Alexander Soifer
    Buy it from: $54.74
    This item will ship on Thursday, April 2.

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    List Price: $99.00
    Publisher: Birkhauser Verlag GmbH
    Binding: Hardcover
    Pages: 190
    Size: 6.50" wide x 9.75" long x 0.75" tall
    Weight: 1.012
    Language: English

    What This Book is About and How it Came into Being
    Ramsey Theory Before Ramsey, Prehistory and Early History: An Essay in 13 Parts
    Overture
    David Hilbert's 1892 Cube Lemma
    The Issai Schur 1916 Theorem
    The Baudet-Schur-Van der Waerden 1927 Theorem
    The Generalized 1928 Schur Theorem
    The Frank Plumpton Ramsey Principle
    The Paul, Gj�rgy, and Esther Happy End Problem
    Richard Rado's Regularity
    Density and Arithmetic Progressions
    The Tibor Gallai Theorem
    De Bruijn-Erdos's 1951 Compactness Theorem
    Khinchin's Small Book of Big Impact
    Long Live the Young Theory!
    References
    Eighty Years of Ramsey R(3, k) � and Counting!
    Basics
    George, Esther, Paul
    Erdos Magic
    An Erdos Gem
    Upper Bounds
    The Lov�sz Local Lemma
    Random Greedy Triangle-Free
    R(3, k) Resolved!
    Random Greedy Triangle-Free Redux
    Epilogue
    References
    Ramsey Numbers Involving Cycles
    Scope and Notation
    Two-Color Numbers Involving Cycles
    Cycles
    Cycles Versus Complete Graphs
    Cycles Versus Wheels
    Cycles Versus Books
    Cycles Versus Other Graphs
    Multicolor Numbers for Cycles
    Three Colors
    More Colors
    Cycles Versus Other Graphs
    Hypergraph Numbers for Cycles
    References
    On the Function of Erdos and Rogers
    Introduction
    The Most Restrictive Case
    Proof of f<sub>s,s+1</sub>(n)&#8804;0(n<sup>1-1/0(s<sup>4</sup>log s)</sup>)[1]
    Proof of �(n<sup>�</sup>)&#8804;f<sub>s,s+1</sub>(n) for s&#8805;2[2]
    Proof of f<sub>s,s+1</sub>(n) &#8804;0(n<sup>2/3</sup>) for s&#8805;2 [4]
    General Bounds
    Proof of �(n<sup>a<sub>k</sub>(s)</sup>)&#8804;f<sub>s,s+k</sub>(n) [14, 15]
    Sketch of the Proof of f<sub>s,s+k</sub>(n)&#8804;0(n<sup>((k+1)/(2k+1))+�</sup>) for s&#8805;s<sub>0</sub> = s<sub>0</sub>(�, k) [4]
    Concluding Remarks
    References
    Large Monochromatic Components in Edge Colorings of Graphs: A Survey
    Introduction
    A Remark of Erdos and Rado and Its Extension
    Colorings from Affine Planes
    Extending Colorings by Substitutions
    2-Colorings
    Type of Spanning Trees, Connectivity, Diameter
    Gallai-Colorings: Substitutions to 2-Colorings
    Multicolorings: Basic Results and Proof Methods
    Complete Bipartite Graphs: Counting Double Stars
    Fractional Transversals: F�redi's Method
    Fine Tuning
    When Both Methods Work: Local Colorings
    Hypergraphs
    Multicolorings: Type of Components
    Components with Large Matching
    Double Stars
    Variations
    Vertex-Coverings by Components
    Coloring by Group Elements
    Coloring Geometric Graphs
    Coloring Noncomplete Graphs
    References
    Szlam's Lemma: Mutant Offspring of a Euclidean Ramsey Problem from 1973, with Numerous Applications
    1973: A Volcano Erupts
    Some Definitions and More Background
    What Happened to the Rather Red Coloring Problem from 1973?
    Distance Graphs
    Szlam's Lemma, a Connection Between Rather Red Colorings and Chromatic Numbers
    van der Waerden Numbers, Cyclic van der Waerden Numbers, and a Lower Bound on Them Both
    References
    Open Problems in Euclidean Ramsey Theory
    Introduction
    Ramsey Sets
    Unit Distance Graphs
    More General Distance Graphs
    References
    Chromatic Number of the Plane & Its Relatives, History, Problems and Results: An Essay in 11 Parts
    The Problem
    The History
    Polychromatic Number of the Plane & Results Near the Lower Bound
    De Bruijn-Erdos Reduction to Finite Sets and Results Near the Lower Bound
    Polychromatic Number of the Plane & Results Near the Upper Bound
    Continum of 6-Colorings of the Plane
    Chromatic Number of the Plane in Special Circumstances
    Colored Space
    Rational Coloring
    Axioms of Set Theory and the Chromatic Number of Graphs
    Predicting the Future
    References
    Euclidean Distance Graphs on the Rational Points
    Definitions
    The Search for �(R<sup>n</sup>, 1) Leads to the Search for �(Q<sup>n</sup>, 1)
    Distances Other Than 1?
    Problems
    References
    Open Problems Session
    Problems Submitted
    Problems Submitted
    Problems on Topological Stability of Chromatic Numbers Submitted
    Problem on the Gallai-Ramsey Structure, Submitted
    Problems Involving Triangles, Submitted
    Problems on Chromatic Number of the Plane and Its Relatives, Submitted

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