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| Softcover ISBN: | 978-1-4704-4071-8 |
| Product Code: | MEMO/263/1274 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 263; 2020; 125 ppMSC: Primary 05; 60
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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Table of Contents
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Chapters
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1. Introduction
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2. An overview of the proof
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3. Martingale bounds: The line of peril and the line of death
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4. Tracking everything else
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5. Tracking $Y_e$, and mixing in the $Y$-graph
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6. Whirlpools and Lyapunov functions
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7. Independent sets and maximum degrees in $G_{n,\triangle }$
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Acknowledgements
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Additional Material
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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.
-
Chapters
-
1. Introduction
-
2. An overview of the proof
-
3. Martingale bounds: The line of peril and the line of death
-
4. Tracking everything else
-
5. Tracking $Y_e$, and mixing in the $Y$-graph
-
6. Whirlpools and Lyapunov functions
-
7. Independent sets and maximum degrees in $G_{n,\triangle }$
-
Acknowledgements
