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008			240724s1920 xx o ||||0 eng d
020			_a9781470456566 _q(electronic bk.)
020			_z9781470440718
035			_a(MiAaPQ)EBC6176737
035			_a(Au-PeEL)EBL6176737
035			_a(OCoLC)1151199549
040			_aMiAaPQ _beng _erda _epn _cMiAaPQ _dMiAaPQ
050		4	_aQA164 _b.P668 2020
082	0		_a511/.6
100	1		_aPontiveros, Gonzalo Fiz.
245	1	4	_aThe Triangle-Free Process and the Ramsey Number _R(3,k) _.
250			_a1st ed.
264		1	_aProvidence : _bAmerican Mathematical Society, _c1920.
264		4	_c©1920.
300			_a1 online resource (138 pages)
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
490	1		_aMemoirs of the American Mathematical Society Series ; _vv.263
505	0		_aCover -- Title page -- Chapter 1. Introduction -- 1.1. Random graph processes -- 1.2. The triangle-free process -- Chapter 2. An overview of the proof -- Chapter 3. Martingale bounds: The line of peril and the line of death -- 3.1. The line of peril and the line of death -- 3.2. A general lemma -- 3.3. The events \X( ), \Y( ), \Z( ) and \Q( ) -- 3.4. Tracking ₑ -- Chapter 4. Tracking everything else -- 4.1. Building sequences -- 4.2. Self-correction -- 4.3. Creating and destroying copies of -- 4.4. Balanced non-tracking graph structures -- 4.5. Bounding the maximum change in *ᵩ( ) -- 4.6. The land before time = -- 4.7. Proof of Theorem 4.1 -- Chapter 5. Tracking ₑ, and mixing in the -graph -- 5.1. Mixing inside open neighbourhoods -- 5.2. Mixing in the whole -graph -- 5.3. Creating and destroying -walks -- 5.4. Self-correction -- 5.5. The Lines of Peril and Death -- Chapter 6. Whirlpools and Lyapunov functions -- 6.1. Whirlpools -- 6.2. Lyapunov functions -- 6.3. The proof of Theorems 2.1, 2.4, 2.5, 2.7 and 2.11 -- Chapter 7. Independent sets and maximum degrees in _{ ,\triangle} -- 7.1. A sketch of the proof -- 7.2. Partitioning the bad events -- 7.3. The events \A( , ) and \A'( , ) -- 7.4. The events \B( , )∩\D( , )^{ } and \B'( , )∩\D( , )^{ } -- 7.5. The events \C( , ) and \C'( , ) -- 7.6. The event \D( , ) -- 7.7. The proof of Propositions 7.1 and 7.2 -- Acknowledgements -- Bibliography -- Back Cover.
520			_aThe areas of Ramsey theory and random graphs have been closely linked ever since Erdős'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.
588			_aDescription based on publisher supplied metadata and other sources.
590			_aElectronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
650		0	_aProbability theory and stochastic processes -- Combinatorial probability.
655		4	_aElectronic books.
700	1		_aGriffiths, Simon.
700	1		_aMorris, Robert.
776	0	8	_iPrint version: _aPontiveros, Gonzalo Fiz _tThe Triangle-Free Process and the Ramsey Number _R(3,k) _dProvidence : American Mathematical Society,c1920 _z9781470440718
797	2		_aProQuest (Firm)
830		0	_aMemoirs of the American Mathematical Society Series
856	4	0	_uhttps://ebookcentral.proquest.com/lib/orpp/detail.action?docID=6176737 _zClick to View
999			_c17666 _d17666