Generalized Ramsey numbers of cycles, paths, and hypergraphs
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909217027260416 |
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| author | Bal, Deepak Bennett, Patrick Heath, Emily Zerbib, Shira |
| author_facet | Bal, Deepak Bennett, Patrick Heath, Emily Zerbib, Shira |
| contents | Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in \mathcal{H}$ in $G$ receives at least $q$ different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\mathcal{H}$ is a set of cycles or paths, and when $G=K_n^k$ and $\mathcal{H}$ contains a clique on $k+2$ vertices or a tight cycle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Ramsey numbers of cycles, paths, and hypergraphs Bal, Deepak Bennett, Patrick Heath, Emily Zerbib, Shira Combinatorics Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in \mathcal{H}$ in $G$ receives at least $q$ different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\mathcal{H}$ is a set of cycles or paths, and when $G=K_n^k$ and $\mathcal{H}$ contains a clique on $k+2$ vertices or a tight cycle. |
| title | Generalized Ramsey numbers of cycles, paths, and hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.15904 |