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The Triangle-Free Process and the Ramsey Number $R(3,k)$, Paperback / softback Book

The Triangle-Free Process and the Ramsey Number $R(3,k)$ Paperback / softback

Part of the Memoirs of the American Mathematical Society series

Paperback / softback

Description

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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