Journal article
Upper bounds for multicolour Ramsey numbers
- Abstract:
-
The $r$-colour Ramsey number $R_r(k)$ is the minimum $n \in \mathbb{N}$ such that every $r$-colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove, for each fixed $r \geqslant 2$, that \[ R_r(k) \leqslant e^{-\delta k_r} r^k \] for some constant $\delta = \delta(r) > 0$ and all sufficiently large $k \in \mathbb{N}$. For each $r \geqslant 3$, this is the first exponential improvement over the upper bound of Erd\H{o}s and Szekeres from 1935. In the case $r = 2$, it gives a different (and significantly shorter) proof of a recent result of Campos, Griffiths, Morris and Sahasrabudhe [Ann. Math., To appear].
- Publication status:
- Published
- Peer review status:
- Peer reviewed
Actions
Access Document
- Files:
-
-
(Preview, Accepted manuscript, pdf, 422.9KB, Terms of use)
-
- Publisher copy:
- 10.1090/jams/1069
Authors
- Funder identifier:
- https://ror.org/050q5pk40
- Grant:
- R-2412-51283
- Funder identifier:
- https://ror.org/0472cxd90
- Grant:
- R-2412-51283
- Funder identifier:
- https://ror.org/03swz6y49
- Grant:
- R-2412-51283
- Funder identifier:
- https://ror.org/03kk0s825
- Grant:
- R-2412-51283
- Publisher:
- American Mathematical Society
- Journal:
- Journal of the American Mathematical Society More from this journal
- Volume:
- 39
- Issue:
- 3
- Pages:
- 765-780
- Publication date:
- 2026-01-16
- DOI:
- EISSN:
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1088-6834
- ISSN:
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0894-0347
- Language:
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English
- Pubs id:
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2367202
- Local pid:
-
pubs:2367202
- Deposit date:
-
2026-04-15
- ARK identifier:
Terms of use
- Copyright holder:
- Balister et al
- Copyright date:
- 2026
- Rights statement:
- © Copyright 2026 by Paul Balister; Béla Bollobás; Marcelo Campos; Simon Griffiths; Eoin Hurley; Robert Morris; Julian Sahasrabudhe; Marius Tiba
- Notes:
- The author accepted manuscript (AAM) of this paper has been made available under the University of Oxford's Open Access Publications Policy, and a CC BY public copyright licence has been applied.
- Licence:
- CC Attribution (CC BY)
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