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The Ramsey number of books

Conlon, David (2019) The Ramsey number of books. . (Unpublished)

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We show that in every two-colouring of the edges of the complete graph K_N there is a monochromatic K_k which can be extended in at least (1 + o_(k)(1))2^(-k)N ways to a monochromatic K_(k+1). This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book B_n^(k) to be the graph consisting of n copies of K_(k+1) all sharing a common K_k, we show that the Ramsey number r(B_n^(k)) = 2^(k)n + o_(k)(n). In this form, our result answers a question of Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Conlon, David0000-0001-5899-1829
Additional Information:Research supported by a Royal Society University Research Fellowship and ERC Starting Grant 676632. This paper was partially written while I was visiting the California Institute of Technology as a Moore Distinguished Scholar and I am extremely grateful for their kind support. I am also much indebted to Jacob Fox, Lisa Sauermann and Yuval Wigderson for pointing out a subtle error in the first version of this paper. Finally, I would like to thank the anonymous reviewers for several helpful remarks which improved the presentation.
Funding AgencyGrant Number
European Research Council (ERC)676632
Gordon and Betty Moore FoundationUNSPECIFIED
Record Number:CaltechAUTHORS:20190819-170924800
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:98030
Deposited By: Melissa Ray
Deposited On:20 Aug 2019 23:12
Last Modified:03 Oct 2019 21:37

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