Ramsey numbers of long even cycles versus books
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916980301234176 |
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| author | Lin, Qizhong Song, Shixi |
| author_facet | Lin, Qizhong Song, Shixi |
| contents | For any positive integers $k$ and $n$, let $B_n^{(k)}$ be the book graph consisting of $n$ copies of the complete graph $K_{k+1}$ sharing a common $K_k$. Let $C_m$ be a cycle of length $m$. Prior work by Allen, Łuczak, Polcyn, and Zhang (2023) established the Ramsey number $R(C_{m},B_n^{(1)})$ for all sufficiently large even integer $m = Ω(n^{9/10})$. Recently, Hu, Lin, Łuczak, Ning, and Peng (2025) obtained the exact value of $R(C_{m},B_n^{(2)})$ under the same asymptotic conditions. A natural problem is to determine the exact value of $R(C_{m},B_n^{(k)})$ for each fixed $k\ge3$ under similar conditions. This paper provides a complete solution to this problem. The lower bound is proved by an explicit construction, while the tight upper bound is established by analyzing the corresponding Ramsey graph using semi-random ideas. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ramsey numbers of long even cycles versus books Lin, Qizhong Song, Shixi Combinatorics For any positive integers $k$ and $n$, let $B_n^{(k)}$ be the book graph consisting of $n$ copies of the complete graph $K_{k+1}$ sharing a common $K_k$. Let $C_m$ be a cycle of length $m$. Prior work by Allen, Łuczak, Polcyn, and Zhang (2023) established the Ramsey number $R(C_{m},B_n^{(1)})$ for all sufficiently large even integer $m = Ω(n^{9/10})$. Recently, Hu, Lin, Łuczak, Ning, and Peng (2025) obtained the exact value of $R(C_{m},B_n^{(2)})$ under the same asymptotic conditions. A natural problem is to determine the exact value of $R(C_{m},B_n^{(k)})$ for each fixed $k\ge3$ under similar conditions. This paper provides a complete solution to this problem. The lower bound is proved by an explicit construction, while the tight upper bound is established by analyzing the corresponding Ramsey graph using semi-random ideas. |
| title | Ramsey numbers of long even cycles versus books |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.26323 |