A new lower bound for the Ramsey numbers $R(3,k)$
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915292800614400 |
|---|---|
| author | Campos, Marcelo Jenssen, Matthew Michelen, Marcus Sahasrabudhe, Julian |
| author_facet | Campos, Marcelo Jenssen, Matthew Michelen, Marcus Sahasrabudhe, Julian |
| contents | We prove a new lower bound for the off-diagonal Ramsey numbers, \[ R(3,k) \geq \bigg( \frac{1}{3}+ o(1) \bigg) \frac{k^2}{\log k }\, , \] thereby narrowing the gap between the upper and lower bounds to a factor of $3+o(1)$. This improves the best known lower bound of $(1/4+o(1))k^2/\log k$ due, independently, to Bohman and Keevash, and Fiz Pontiveros, Griffiths and Morris, resulting from their celebrated analysis of the triangle-free process. As a consequence, we disprove a conjecture of Fiz Pontiveros, Griffiths and Morris that the constant $1/4$ is sharp. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new lower bound for the Ramsey numbers $R(3,k)$ Campos, Marcelo Jenssen, Matthew Michelen, Marcus Sahasrabudhe, Julian Combinatorics Probability We prove a new lower bound for the off-diagonal Ramsey numbers, \[ R(3,k) \geq \bigg( \frac{1}{3}+ o(1) \bigg) \frac{k^2}{\log k }\, , \] thereby narrowing the gap between the upper and lower bounds to a factor of $3+o(1)$. This improves the best known lower bound of $(1/4+o(1))k^2/\log k$ due, independently, to Bohman and Keevash, and Fiz Pontiveros, Griffiths and Morris, resulting from their celebrated analysis of the triangle-free process. As a consequence, we disprove a conjecture of Fiz Pontiveros, Griffiths and Morris that the constant $1/4$ is sharp. |
| title | A new lower bound for the Ramsey numbers $R(3,k)$ |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2505.13371 |