An exponential improvement for diagonal Ramsey
Abstract
The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that \[ R(k) \leqslant (4 - \varepsilon)^k \] for some constant $\varepsilon > 0$. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.
Fields of Study & Subjects
Limits and Structures in Graph Theory
Advanced Topology and Set Theory
Advanced Graph Theory Research
Publication details
- Journal / source
- Annals of Mathematics
- Publisher
- Annals of Mathematics
- Volume
- 203
- Issue
- 3
- Publication date
- 1 May 2026
- Language
- EN