The Triangle-Free Process and the Ramsey Number

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.

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 fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3,k) = \Theta \big ( k^2 / \log k \big ). In this paper the authors improve the results of both Bohman and Kim and follow the triangle-free process all the way to its asymptotic end.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.