Upper bounds for multicolour Ramsey numbers
Abstract
The r r -colour Ramsey number R r ( k ) R_r(k) is the minimum n ∈ N n \in \mathbb {N} such that every r r -colouring of the edges of the complete graph K n K_n on n n vertices contains a monochromatic copy of K k K_k . We prove, for each fixed r ⩾ 2 r \geqslant 2 , that R r ( k ) ⩽ e − δ k r r k \begin{equation*} R_r(k) \leqslant e^{-\delta k} r^{rk} \end{equation*} for some constant δ = δ ( r ) > 0 \delta = \delta (r) > 0 and all sufficiently large k ∈ N k \in \mathbb {N} . For each r
Fields of Study & Subjects
Limits and Structures in Graph Theory
Markov Chains and Monte Carlo Methods
Advanced Topology and Set Theory
Publication details
- Journal / source
- Journal of the American Mathematical Society
- Publisher
- American Mathematical Society (AMS)
- Volume
- 39
- Issue
- 3
- Pages
- 765–780
- Publication date
- 12 January 2026
- Language
- EN