## Abstract

The Ramsey number r(C

We prove that, for some absolute constant C ≥ 1, we have r(C

This proves the conjecture of Erdős, 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 Erdős, Faudree, Rousseau, and Schelp.

_{ℓ},K_{n}) 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, Erdős, Faudree, Rousseau, and Schelp conjectured that r(C_{ℓ},K_{n}) = (ℓ−1)(n−1)+1 for ℓ ≥ n ≥ 3 provided (ℓ, n) ≠ (3,3).We prove that, for some absolute constant C ≥ 1, we have r(C

_{ℓ},K_{n}) = (ℓ−1)(n−1)+1 provided ℓ ≥ C (log n / log log n). Up to the value of C this is tight since we also show that, for any ε > 0 and n > n_{0}(ε), we have r(C_{ℓ},K_{n}) ≫ (ℓ−1)(n−1)+1 for all 3 ≤ ℓ ≤ (1−ε)(log n / log log n).This proves the conjecture of Erdős, 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 Erdős, Faudree, Rousseau, and Schelp.

Original language | English |
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Article number | rnz119 |

Pages (from-to) | 275-300 |

Number of pages | 26 |

Journal | International Mathematics Research Notices |

Volume | 2021 |

Issue number | 1 |

Early online date | 10 Jul 2019 |

DOIs | |

Publication status | E-pub ahead of print - 10 Jul 2019 |

## Keywords

- Ramsey theory