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The Triangle-Free Process and the Ramsey Number R(3,k)

The areas of Ramsey theory and random graphs have been closely linked ever since Erdős's famous proof in 1947 that the ""diagonal"" Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially p...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Pontiveros, Gonzalo Fiz
Otros Autores: Griffiths, Simon, Morris, Robert
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Providence : American Mathematical Society, 1920.
Colección:Memoirs of the American Mathematical Society Ser.
Temas:
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Cover
  • Title page
  • Chapter 1. Introduction
  • 1.1. Random graph processes
  • 1.2. The triangle-free process
  • Chapter 2. An overview of the proof
  • Chapter 3. Martingale bounds: The line of peril and the line of death
  • 3.1. The line of peril and the line of death
  • 3.2. A general lemma
  • 3.3. The events \X( ), \Y( ), \Z( ) and \Q( )
  • 3.4. Tracking ₑ
  • Chapter 4. Tracking everything else
  • 4.1. Building sequences
  • 4.2. Self-correction
  • 4.3. Creating and destroying copies of
  • 4.4. Balanced non-tracking graph structures
  • 4.5. Bounding the maximum change in *ᵩ( )
  • 4.6. The land before time =
  • 4.7. Proof of Theorem 4.1
  • Chapter 5. Tracking ₑ, and mixing in the -graph
  • 5.1. Mixing inside open neighbourhoods
  • 5.2. Mixing in the whole -graph
  • 5.3. Creating and destroying -walks
  • 5.4. Self-correction
  • 5.5. The Lines of Peril and Death
  • Chapter 6. Whirlpools and Lyapunov functions
  • 6.1. Whirlpools
  • 6.2. Lyapunov functions
  • 6.3. The proof of Theorems 2.1, 2.4, 2.5, 2.7 and 2.11
  • Chapter 7. Independent sets and maximum degrees in _{ ,\triangle}
  • 7.1. A sketch of the proof
  • 7.2. Partitioning the bad events
  • 7.3. The events \A( , ) and \A'( , )
  • 7.4. The events \B( , )∩\D( , )^{ } and \B'( , )∩\D( , )^{ }
  • 7.5. The events \C( , ) and \C'( , )
  • 7.6. The event \D( , )
  • 7.7. The proof of Propositions 7.1 and 7.2
  • Acknowledgements
  • Bibliography
  • Back Cover