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PreprintCC BY 4.0arXiv

An Improved Upper Bound for Multicolour Ramsey Numbers

  • Sunghyeon Jo
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Abstract

Let Rr(k) denote the diagonal r-colour Ramsey number. We prove that there exist absolute constants c,K>0 such that Rr(k)≤ rrkexp (−ck/rlog2(2r)) for every r≥2 and every k≥ Kr2log6(2r). This improves the exponential saving in a recent bound of Yang and Mao by a factor of order rlog2(2r). The proof proceeds through an off-diagonal bound, which asymptotically improves the classical multinomial bound throughout a neighbourhood of the diagonal.

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Cite

BibTeXRIS
BibTeX
@misc{jo2026improved,
  title = {{An Improved Upper Bound for Multicolour Ramsey Numbers}},
  author = {Sunghyeon Jo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04596},
  eprint = {2609.04596},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04596},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - An Improved Upper Bound for Multicolour Ramsey Numbers
AU  - Sunghyeon Jo
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.04596
UR  - https://arxiv.org/abs/2609.04596
AB  - Let R_r(k) denote the diagonal r-colour Ramsey number. We prove that there exist absolute constants c,K>0 such that R_r(k)≤ r^rkexp (−ck/rlog^2(2r)) for every r≥2 and every k≥ Kr^2log^6(2r). This improves the exponential saving in a recent bound of Yang and Mao by a factor of order rlog^2(2r). The proof proceeds through an off-diagonal bound, which asymptotically improves the classical multinomial bound throughout a neighbourhood of the diagonal.
ER  -