An exponential improvement for diagonal Ramsey

By Marcelo Campos, Simon Griffiths, Robert Morris, Julian Sahasrabudhe. 2023

99 citations recorded by the source corpus.

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 Erd\H{o}s and Szekeres, proved in 1935.

View Paper: https://www.semanticscholar.org/paper/839ddc8ef3d24c4cc389275d8691224b128f65fc

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Citations in the corpus

• The asymptotics of r(4,t) (2023, background): “The quantities r ( t, t ) are sometimes referred to as diagonal Ramsey Numbers – see Campos, Griffiths, Morris, Sahasrabudhe [12] for a recent major breakthrough – whereas r ( s, t ) for fixed s and t → ∞ are referred to as off-diagonal Ramsey numbers .”