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

ISBN-13: 9781470440718

Gonzalo Fiz Pontiveros;Simon Griffiths;Robert Morris

Opis

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, riangle $. 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 $R(3,k) = Theta ig ( k^2 / log k ig )$. 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.

Spis treści

• Introduction
• An overview of the proof
• Martingale bounds: the line of peril and the line of death
• Tracking everything else
• Tracking $Y_e$, and mixing in the $Y$-graph
• Whirlpools and Lyapunov functions
• Independent sets and maximum degrees in $G_n,\triangle $
• Bibliography.