The Triangle-Free Process and the Ramsey Number $R(3,k)$ PDF
by Gonzalo Fiz Pontiveros
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The areas of Ramsey theory and random graphs have been closely linked ever since Erdos'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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- Format:PDF
- Publisher:American Mathematical Society
- Publication Date:04/03/2020
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- ISBN:9781470456566
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Information
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Download Now
- Format:PDF
- Publisher:American Mathematical Society
- Publication Date:04/03/2020
- Category:
- ISBN:9781470456566