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PreprintarXiv licencearXiv

On the multicolour Ramsey numbers R(3,3,k)

  • Bruno Andrades
  • Marcelo Campos
  • Robert Morris
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Abstract

In this paper we determine the Ramsey number R(3,3,k) up to a constant factor, showing that R(3,3,k) = Θ( k3/((log k)2) ). The proof of the lower bound combines the Hefty-Horn-King-Pfender construction for R(3,k) with the method of Alon and Rödl. Using the same proof, we also determine the r-colour Ramsey numbers Rr(3,…,3,k) up to a constant factor for every fixed r geqslant 3.

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Cite

BibTeXRIS
BibTeX
@misc{andrades2026multicolour,
  title = {{On the multicolour Ramsey numbers $R(3,3,k)$}},
  author = {Bruno Andrades and Marcelo Campos and Robert Morris},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28455},
  eprint = {2609.28455},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28455},
}
RIS
TY  - UNPB
TI  - On the multicolour Ramsey numbers R(3,3,k)
AU  - Bruno Andrades
AU  - Marcelo Campos
AU  - Robert Morris
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28455
UR  - https://arxiv.org/abs/2609.28455
AB  - In this paper we determine the Ramsey number R(3,3,k) up to a constant factor, showing that R(3,3,k) = Θ( k^3/((log k)^2) ). The proof of the lower bound combines the Hefty-Horn-King-Pfender construction for R(3,k) with the method of Alon and Rödl. Using the same proof, we also determine the r-colour Ramsey numbers R_r(3,…,3,k) up to a constant factor for every fixed r geqslant 3.
ER  -