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Cycle-complete ramsey numbers

Keevash, Peter, Long, Eoin and Skokan, Jozef (2019) Cycle-complete ramsey numbers. International Mathematics Research Notices.

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Identification Number: 10.1093/imrn/rnz119


The Ramsey numberr(Cℓ, Kn) is the smallest natural numberNsuch that every red/blueedge-colouring of a clique of orderNcontains a red cycle of lengthℓor a blue clique of ordern.In 1978, Erd ̋os, Faudree, Rousseau and Schelp conjectured thatr(Cℓ, Kn) = (ℓ−1)(n−1) + 1 forℓ≥n≥3 provided (ℓ, n)6= (3,3).We prove that, for some absolute constantC≥1, we haver(Cℓ, Kn) = (ℓ−1)(n−1) + 1providedℓ≥Clognlog logn. Up to the value ofCthis is tight since we also show that, for anyε >0andn > n0(ε), we haver(Cℓ, Kn)≫(ℓ−1)(n−1) + 1 for all 3≤ℓ≤(1−ε)lognlog logn.This proves the conjecture of Erd ̋os, Faudree, Rousseau and Schelp for largeℓ, a stronger form ofthe conjecture due to Nikiforov, and answers (up to multiplicative constants) two further questionsof Erd ̋os, Faudree, Rousseau and Schelp.

Item Type: Article
Official URL:
Additional Information: © 2019 Oxford University Press
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 29 May 2019 15:00
Last Modified: 27 Oct 2020 00:05

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