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Exact Ramsey numbers of odd cycles via nonlinear optimisation

Jenssen, Matthew and Skokan, Jozef ORCID: 0000-0003-3996-7676 (2021) Exact Ramsey numbers of odd cycles via nonlinear optimisation. Advances in Mathematics, 376. ISSN 0001-8708

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Identification Number: 10.1016/j.aim.2020.107444


For a graph G, the k-colour Ramsey number R k(G) is the least integer N such that every k-colouring of the edges of the complete graph K N contains a monochromatic copy of G. Let C n denote the cycle on n vertices. We show that for fixed k≥2 and n odd and sufficiently large, R k(C n)=2 k−1(n−1)+1. This resolves a conjecture of Bondy and Erdős for large n. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal k-colourings for this problem and perfect matchings in the k-dimensional hypercube Q k.

Item Type: Article
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Additional Information: © 2020 Elsevier Inc
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 05 Oct 2020 09:18
Last Modified: 06 Jun 2024 22:57

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