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The Triangle-Free Process and the Ramsey Number R(3, K)

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The Triangle-Free Process and the Ramsey Number R(3, K) Synopsis

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.

About This Edition

ISBN: 9781470440718
Publication date:
Author: Gonzalo Fiz Pontiveros
Publisher: American Mathematical Society
Format: Paperback
Pagination: 125 pages
Series: Memoirs of the American Mathematical Society
Genres: Discrete mathematics
Probability and statistics
Combinatorics and graph theory