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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.09521 |
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| _version_ | 1866908478142939136 |
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| author | Campos, Marcelo Griffiths, Simon Morris, Robert Sahasrabudhe, Julian |
| author_facet | Campos, Marcelo Griffiths, Simon Morris, Robert Sahasrabudhe, Julian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_09521 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An exponential improvement for diagonal Ramsey Campos, Marcelo Griffiths, Simon Morris, Robert Sahasrabudhe, Julian Combinatorics 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. |
| title | An exponential improvement for diagonal Ramsey |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2303.09521 |