Keevash, Peter, Long, Eoin and Skokan, Jozef
(2021)
*Cycle-complete ramsey numbers.*
International Mathematics Research Notices, 2021 (1).
275 – 300.
ISSN 1073-7928

Text (Cycle-complete)
- Accepted Version
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## Abstract

The Ramsey number r(Cℓ, Kn) is the smallest natural number N such that every red/blue edge-colouring of a clique of order N contains a red cycle of length ℓ or a blue clique of order n. In 1978, Erdos, Faudree, Rousseau and Schelp conjectured that r(Cℓ, Kn) = (ℓ − 1)(n − 1) + 1 for ℓ ≥ n ≥ 3 provided (ℓ, n) 6= (3, 3). We prove that, for some absolute constant C ≥ 1, we have r(Cℓ, Kn) = (ℓ − 1)(n − 1) + 1 provided ℓ ≥ C logloglognn. Up to the value of C this is tight since we also show that, for any ε > 0 and n > n0(ε), we have r(Cℓ, Kn) ≫ (ℓ − 1)(n − 1) + 1 for all 3 ≤ ℓ ≤ (1 − ε)logloglognn. This proves the conjecture of Erdos, Faudree, Rousseau and Schelp for large ℓ, a stronger form of the conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questions of Erdos, Faudree, Rousseau and Schelp.

Item Type: | Article |
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Official URL: | https://academic.oup.com/imrn |

Additional Information: | © 2019 The Authors |

Divisions: | Mathematics |

Subjects: | Q Science > QA Mathematics |

Date Deposited: | 29 May 2019 15:00 |

Last Modified: | 26 Jul 2021 23:12 |

URI: | http://eprints.lse.ac.uk/id/eprint/100791 |

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