The asymptotic of off-diagonal online Ramsey numbers for paths

We prove that for every k≥10, the online Ramsey number for paths Pk and Pn satisfies r̃(Pk,Pn)≥53n+k9−4, matching up to a linear term in k the upper bound recently obtained by Bednarska-Bzdęga (2024). In particular, this implies limn→∞r̃(Pk,Pn)n=53, whenever 10≤k=o(n), disproving a conjecture by Cym...

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Published in:European journal of combinatorics 2024-12, Vol.122, p.104032, Article 104032
Main Authors: Mond, Adva, Portier, Julien
Format: Article
Language:English
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Summary:We prove that for every k≥10, the online Ramsey number for paths Pk and Pn satisfies r̃(Pk,Pn)≥53n+k9−4, matching up to a linear term in k the upper bound recently obtained by Bednarska-Bzdęga (2024). In particular, this implies limn→∞r̃(Pk,Pn)n=53, whenever 10≤k=o(n), disproving a conjecture by Cyman et al. (2015).
ISSN:0195-6698
DOI:10.1016/j.ejc.2024.104032