Abstract
We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant $C$ such that \[r(k+1, k+1) \leq k^{- C {\log k}/{\log \log k}} \textstyle \binom{2k}{k}.\]
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[CGW89] F. R. K. Chung, R. L. Graham, and R. M. Wilson, "Quasi-random graphs," Combinatorica, vol. 9, iss. 4, pp. 345-362, 1989.
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AUTHOR = {Chung, F. R. K. and Graham, R. L. and Wilson, R. M.},
TITLE = {Quasi-random graphs},
JOURNAL = {Combinatorica},
FJOURNAL = {Combinatorica. An International Journal on Combinatorics and the Theory of Computing},
VOLUME = {9},
YEAR = {1989},
NUMBER = {4},
PAGES = {345--362},
ISSN = {0209-9683},
CODEN = {COMBDI},
MRCLASS = {05C80},
MRNUMBER = {91e:05074},
ZBLNUMBER = {0715.05057},
MRREVIEWER = {Zbigniew Palka},
} -
[ES35] P. ErdHos and G. Szekeres, "A combinatorial problem in geometry," Compositio Math., vol. 2, pp. 463-470, 1935.
@article{ES35, MRKEY = {1556929},
AUTHOR = {Erdős, P. and Szekeres, G.},
TITLE = {A combinatorial problem in geometry},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {2},
YEAR = {1935},
PAGES = {463--470},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {Contributed Item},
MRNUMBER = {1556929},
ZBLNUMBER = {0012.27010},
JFMNUMBER = {61.0651.04},
} -
[GR87] R. L. Graham and V. Rödl, "Numbers in Ramsey theory," in Surveys in Combinatorics 1987, Whitehead, C., Ed., Cambridge: Cambridge Univ. Press, 1987, pp. 111-153.
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AUTHOR = {Graham, R. L. and R{ö}dl, V.},
EDITOR = {Whitehead, C.},
TITLE = {Numbers in {R}amsey theory},
BOOKTITLE = {Surveys in Combinatorics 1987},
VENUE = {New Cross, 1987},
SERIES = {London Math. Soc. Lecture Note Ser.},
NUMBER = {123},
PAGES = {111--153},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1987},
MRCLASS = {05C55},
MRNUMBER = {89b:05125},
ZBLNUMBER = {0633.05049},
MRREVIEWER = {Walter A. Deuber},
} -
[R30] F. P. Ramsey, "On a problem of formal logic," Proc. London Math. Soc., vol. 30, pp. 264-286, 1929.
@article{R30,
author = {Ramsey, F. P.},
TITLE = {On a problem of formal logic},
JOURNAL = {Proc. London Math. Soc.},
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VOLUME = {30},
YEAR = {1929},
NUMBER = {},
PAGES = {264--286},
ISSN = {},
JFMNUMBER = {55.0032.04},
} -
[T87] A. Thomason, "Pseudorandom graphs," in Random Graphs ’85, Karoński Michałand Palka, Z., Ed., Amsterdam: North-Holland, 1987, pp. 307-331.
@incollection{T87, MRKEY = {930498},
AUTHOR = {Thomason, Andrew},
EDITOR = {Karo{ń}ski, Micha{\l}and Palka, Zbigniew},
TITLE = {Pseudorandom graphs},
BOOKTITLE = {Random Graphs {\rm'85}},
VENUE = {Poznań, Poland, 1985},
SERIES = {North-Holland Math. Stud.},
NUMBER = {144},
PAGES = {307--331},
PUBLISHER = {North-Holland},
ADDRESS = {Amsterdam},
YEAR = {1987},
MRCLASS = {05C80},
MRNUMBER = {89d:05158},
MRREVIEWER = {Edward R. Scheinerman},
ZBLNUMBER = {0632.05045},
} -
[T88] A. Thomason, "An upper bound for some Ramsey numbers," J. Graph Theory, vol. 12, iss. 4, pp. 509-517, 1988.
@article{T88, MRKEY = {968746},
AUTHOR = {Thomason, Andrew},
TITLE = {An upper bound for some {R}amsey numbers},
JOURNAL = {J. Graph Theory},
FJOURNAL = {Journal of Graph Theory},
VOLUME = {12},
YEAR = {1988},
NUMBER = {4},
PAGES = {509--517},
ISSN = {0364-9024},
CODEN = {JGTHDO},
MRCLASS = {05C55},
MRNUMBER = {90c:05152},
ZBLNUMBER = {0661.05043},
MRREVIEWER = {R. H. Schelp},
}
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