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Book Cover
E-book
Author Pontiveros, Gonzalo Fiz

Title The Triangle-Free Process and the Ramsey Number R(3,k)
Published Providence : American Mathematical Society, 1920

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Description 1 online resource (138 p.)
Series Memoirs of the American Mathematical Society Ser. ; v.263
Memoirs of the American Mathematical Society Ser
Contents 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
Summary 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 provide good lower bounds for the ""off-diagonal"" Ramsey numbers R(3,k). In this model, edges of K_n are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted G_n,\triangle . In 2009, Bohman succeeded in following this process for a positive fra
Notes Description based upon print version of record
Subject Ramsey theory.
Combinatorial analysis.
Combinatorial analysis
Ramsey theory
Probability theory and stochastic processes {For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX} -- Combinatorial probability -- Combinatorial probability.
Combinatorics {For finite fields, see 11Txx} -- Extremal combinatorics -- Ramsey theory [See also 05C55].
Combinatorics {For finite fields, see 11Txx} -- Extremal combinatorics -- Probabilistic methods.
Form Electronic book
Author Griffiths, Simon
Morris, Robert
ISBN 9781470456566
1470456567