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.
ISBN: | 9781470440718 |
Publication date: | 30th April 2020 |
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 |