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The Triangle-Free Process and the Ramsey Number $R(3,k)$ (Memoirs of the American Mathematical Society)

Combinatorics & Graph Theory

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About the Author Gonzalo Fiz Pontiveros, Instituto Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, BrasilSimon Griffiths, Instituto Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, BrasilRobert Morris, Instituto Nacional de Matematica Pura e Aplicada (IMPA), Rio de Janeiro, Brasil Product 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.

Product Specifications

Format: paperback
ASIN: 1470440717
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Combinatorics & Graph Theory
Domain: Amazon UK
Release Date: 30 January 2020
Listed Since: 18 December 2019

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The Triangle-Free Process and the Ramsey Number $R(3,k)$ (Memoirs of the American Mathematical Society)

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