The asymptotics of $r(4,t)$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866916132156342272 |
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| author | Mattheus, Sam Verstraete, Jacques |
| author_facet | Mattheus, Sam Verstraete, Jacques |
| contents | For integers $s,t \geq 2$, the Ramsey numbers $r(s,t)$ denote the minimum $N$ such that every $N$-vertex graph contains either a clique of order $s$ or an independent set of order $t$. In this paper we prove \[ r(4,t) = Ω\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a conjecture of Erdős. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04007 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The asymptotics of $r(4,t)$ Mattheus, Sam Verstraete, Jacques Combinatorics For integers $s,t \geq 2$, the Ramsey numbers $r(s,t)$ denote the minimum $N$ such that every $N$-vertex graph contains either a clique of order $s$ or an independent set of order $t$. In this paper we prove \[ r(4,t) = Ω\Bigl(\frac{t^3}{\log^4 \! t}\Bigr) \quad \quad \mbox{ as }t \rightarrow \infty\] which determines $r(4,t)$ up to a factor of order $\log^2 \! t$, and solves a conjecture of Erdős. |
| title | The asymptotics of $r(4,t)$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2306.04007 |