The Triangle-Free Process and the Ramsey Number $R(3,k)$
Robert Morris author Simon Griffiths author Gonzalo Fiz Pontiveros author
Format:Paperback
Publisher:American Mathematical Society
Published:30th Apr '20
Should be back in stock very soon
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
Dimensions: unknown
Weight: 254g
125 pages