Keevash, Peter, Long, Eoin and Skokan, Jozef ORCID: 0000-0003-3996-7676
(2021)
Cycle-complete ramsey numbers.
International Mathematics Research Notices, 2021 (1).
275 – 300.
ISSN 1073-7928
![]() |
Text (Cycle-complete)
- Accepted Version
Download (294kB) |
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 |
---|---|
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: | 16 Aug 2024 01:41 |
URI: | http://eprints.lse.ac.uk/id/eprint/100791 |
Actions (login required)
![]() |
View Item |