An exponential improvement for Ramsey lower bounds
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866908992219906048 |
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| author | Ma, Jie Shen, Wujie Xie, Shengjie |
| author_facet | Ma, Jie Shen, Wujie Xie, Shengjie |
| contents | We prove a new lower bound on the Ramsey number $r(\ell, C\ell)$ for any constant $C > 1$ and sufficiently large $\ell$, showing that there exists $\varepsilon=\varepsilon(C)> 0$ such that \[ r(\ell, C\ell) \geq \left(p_C^{-1/2} + \varepsilon\right)^\ell, \] where $p_C \in (0, 1/2)$ is the unique solution to $C = \frac{\log p_C}{\log(1 - p_C)}$. This provides the first exponential improvement over the classical lower bound obtained by Erdős in 1947. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12926 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An exponential improvement for Ramsey lower bounds Ma, Jie Shen, Wujie Xie, Shengjie Combinatorics We prove a new lower bound on the Ramsey number $r(\ell, C\ell)$ for any constant $C > 1$ and sufficiently large $\ell$, showing that there exists $\varepsilon=\varepsilon(C)> 0$ such that \[ r(\ell, C\ell) \geq \left(p_C^{-1/2} + \varepsilon\right)^\ell, \] where $p_C \in (0, 1/2)$ is the unique solution to $C = \frac{\log p_C}{\log(1 - p_C)}$. This provides the first exponential improvement over the classical lower bound obtained by Erdős in 1947. |
| title | An exponential improvement for Ramsey lower bounds |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.12926 |