3-uniform monotone paths and multicolor Ramsey numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910712334385152 |
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| author | Suk, Andrew Zeng, Ji |
| author_facet | Suk, Andrew Zeng, Ji |
| contents | The monotone path $P_{n+2}$ is an ordered 3-uniform hypergraph whose vertex set has size $n+2$ and edge set consists of all consecutive triples. In this note, we consider the collection $\mathcal{J}_n$ of ordered 3-uniform hypergraphs named monotone paths with $n$ jumps, and we prove the following relation \begin{equation*} r(3;n) \leq R(P_{n+2},\mathcal{J}_n) \leq 4^n \cdot r(3;n), \end{equation*} where $r(3;n)$ is the multicolor Ramsey number for triangles and $R(P_{n+2},\mathcal{J}_n)$ is the hypergraph Ramsey number for $P_{n+2}$ versus any member of $\mathcal{J}_n$. In particular, whether $r(3;n)$ is exponential, which is a very old problem of Erdős, is equivalent to whether $R(P_{n+2},\mathcal{J}_n)$ is exponential. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_15649 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 3-uniform monotone paths and multicolor Ramsey numbers Suk, Andrew Zeng, Ji Combinatorics The monotone path $P_{n+2}$ is an ordered 3-uniform hypergraph whose vertex set has size $n+2$ and edge set consists of all consecutive triples. In this note, we consider the collection $\mathcal{J}_n$ of ordered 3-uniform hypergraphs named monotone paths with $n$ jumps, and we prove the following relation \begin{equation*} r(3;n) \leq R(P_{n+2},\mathcal{J}_n) \leq 4^n \cdot r(3;n), \end{equation*} where $r(3;n)$ is the multicolor Ramsey number for triangles and $R(P_{n+2},\mathcal{J}_n)$ is the hypergraph Ramsey number for $P_{n+2}$ versus any member of $\mathcal{J}_n$. In particular, whether $r(3;n)$ is exponential, which is a very old problem of Erdős, is equivalent to whether $R(P_{n+2},\mathcal{J}_n)$ is exponential. |
| title | 3-uniform monotone paths and multicolor Ramsey numbers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2411.15649 |