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# An exponential improvement for diagonal Ramsey

Mar 2023

The Ramsey number $R(k)$ is the minimum $n \in \mathbb{N}$ such that everyred-blue colouring of the edges of the complete graph $K_n$ on $n$ verticescontains a monochromatic copy of $K_k$. We prove that $R(k) \leqslant (4 -\varepsilon)^k$ for some constant $\varepsilon > 0$. This is the firstexponential improvement over the upper bound of Erd\H{o}s and Szekeres, provedin 1935.

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