Cycle-complete ramsey numbers

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

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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
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:	12 Dec 2024 01:45
URI:	http://eprints.lse.ac.uk/id/eprint/100791

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